Super hard system of equations
Solve the system of equation for real numbers
begin{split}
(a+b) &(c+d) &= 1 & qquad (1)\
(a^2+b^2)&(c^2+d^2) &= 9 & qquad (2)\
(a^3+b^3)&(c^3+d^3) &= 7 & qquad (3)\
(a^4+b^4)&(c^4+d^4) &=25 & qquad (4)\
end{split}
First I used the identity
$$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(bc+ad)^2$$
Use this identity to (4) too
and simplify (3),
we obtain $$(a^2+b^2-ab)(c^2+d^2-cd)=7$$
And suppose $x=abcd$ use $ac=x /bd , bc=x/ad$
But got stuck...
algebra-precalculus systems-of-equations symmetric-polynomials
add a comment |
Solve the system of equation for real numbers
begin{split}
(a+b) &(c+d) &= 1 & qquad (1)\
(a^2+b^2)&(c^2+d^2) &= 9 & qquad (2)\
(a^3+b^3)&(c^3+d^3) &= 7 & qquad (3)\
(a^4+b^4)&(c^4+d^4) &=25 & qquad (4)\
end{split}
First I used the identity
$$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(bc+ad)^2$$
Use this identity to (4) too
and simplify (3),
we obtain $$(a^2+b^2-ab)(c^2+d^2-cd)=7$$
And suppose $x=abcd$ use $ac=x /bd , bc=x/ad$
But got stuck...
algebra-precalculus systems-of-equations symmetric-polynomials
2
I would use that $$a^3+b^3=(a+b)(a^2+b^2-ab)$$ etc
– Dr. Sonnhard Graubner
yesterday
1
Your system is a very interesting (and puzzling!). Could you say where you found it ?
– Jean Marie
14 hours ago
add a comment |
Solve the system of equation for real numbers
begin{split}
(a+b) &(c+d) &= 1 & qquad (1)\
(a^2+b^2)&(c^2+d^2) &= 9 & qquad (2)\
(a^3+b^3)&(c^3+d^3) &= 7 & qquad (3)\
(a^4+b^4)&(c^4+d^4) &=25 & qquad (4)\
end{split}
First I used the identity
$$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(bc+ad)^2$$
Use this identity to (4) too
and simplify (3),
we obtain $$(a^2+b^2-ab)(c^2+d^2-cd)=7$$
And suppose $x=abcd$ use $ac=x /bd , bc=x/ad$
But got stuck...
algebra-precalculus systems-of-equations symmetric-polynomials
Solve the system of equation for real numbers
begin{split}
(a+b) &(c+d) &= 1 & qquad (1)\
(a^2+b^2)&(c^2+d^2) &= 9 & qquad (2)\
(a^3+b^3)&(c^3+d^3) &= 7 & qquad (3)\
(a^4+b^4)&(c^4+d^4) &=25 & qquad (4)\
end{split}
First I used the identity
$$(a^2+b^2)(c^2+d^2)=(ac-bd)^2+(bc+ad)^2$$
Use this identity to (4) too
and simplify (3),
we obtain $$(a^2+b^2-ab)(c^2+d^2-cd)=7$$
And suppose $x=abcd$ use $ac=x /bd , bc=x/ad$
But got stuck...
algebra-precalculus systems-of-equations symmetric-polynomials
algebra-precalculus systems-of-equations symmetric-polynomials
edited 11 hours ago
Harry Peter
5,46111439
5,46111439
asked yesterday
HeartHeart
1286
1286
2
I would use that $$a^3+b^3=(a+b)(a^2+b^2-ab)$$ etc
– Dr. Sonnhard Graubner
yesterday
1
Your system is a very interesting (and puzzling!). Could you say where you found it ?
– Jean Marie
14 hours ago
add a comment |
2
I would use that $$a^3+b^3=(a+b)(a^2+b^2-ab)$$ etc
– Dr. Sonnhard Graubner
yesterday
1
Your system is a very interesting (and puzzling!). Could you say where you found it ?
– Jean Marie
14 hours ago
2
2
I would use that $$a^3+b^3=(a+b)(a^2+b^2-ab)$$ etc
– Dr. Sonnhard Graubner
yesterday
I would use that $$a^3+b^3=(a+b)(a^2+b^2-ab)$$ etc
– Dr. Sonnhard Graubner
yesterday
1
1
Your system is a very interesting (and puzzling!). Could you say where you found it ?
– Jean Marie
14 hours ago
Your system is a very interesting (and puzzling!). Could you say where you found it ?
– Jean Marie
14 hours ago
add a comment |
4 Answers
4
active
oldest
votes
Hint: $ac=x, bc=y, ad=u, bd=v$, then the equations are
$x+y+u+v=1$
$x^2+y^2+u^2+v^2=9$
$x^3+y^3+u^3+v^3=7$
$x^4+y^4+u^4+v^4=25$
Use Newton-Girard to compute the elementary polynomials.
Then you have the polynomial $P(z)= (z-x)(z-y)(z-u)(z-v)$ with variable $z$.
Solve the quartic equation $P(z)=0$, and there you have the values $x,y,u,v$ in some order.
Note that not in any order: $xv=yu$ must be true, see the definition of these variables.
Of course, once you have $x,y,u,v$, it is easy to compute $a,b,c,d$.
P.S. By this way we can get:
$${x,y,u,v}={-1,2,sqrt2,-sqrt2},$$ which gives $abcd=-2.$
Up to symmetry, the solution is $(a,b,c,d)= (t, -sqrt{2}t, -frac{1}{t}, -frac{sqrt{2}}{t})$ for any $tneq 0$.
(By up to symmetry, I mean you can switch $a$ and $b$, you can switch $c$ and $d$, and you can switch the pair $(a,b)$ with $(c,d)$, so there are $8$ symmetries.)
2
I solved your system. What is the rest?
– Michael Rozenberg
yesterday
2
Yes, I see now. We can kill it.
– Michael Rozenberg
yesterday
I am sorry but I don't find your derivation very clear. Besides, a) the solution you propose doesn't work : from the four equations $(1) ac=-1=x, (2) bc=2=y, (3) ad=sqrt{2}=u, (4) bd=-sqrt{2}=v$, if I divide (1) by (3), I get $c/d=-1/sqrt{2}$, whereas, if I divide (2) by (3), I get $c/d=sqrt{2}$, which is contradictory. b) See the infinite family of solutions provided by Claude.
– Jean Marie
yesterday
1
@JeanMarie I am sorry if you don't find it clear, but could you be more specific? I checked, and it seems to me that the solution (given by @MichaelRozenberg) is correct for $x,y,u,v$. From that it should be a routine taks to solve the original system of equations. You made a mistake though: as I said, you cannot randomly pick which root is which variable. It should be $ab=-1, cd=2, ac=sqrt{2}, bd=-sqrt{2}$, or one of the other seven symmetrical choices. Keep in mind that $xv=yu$.
– A. Pongrácz
yesterday
I realize that I have misunderstood your set equality $${x,y,u,v}={-1,2,sqrt2,-sqrt2}$$ that I have interpreted as a t-uple equality $$(x,y,u,v)=(-1,2,sqrt2,-sqrt2)$$... But what I find not evident to understand is a) the (clever !) transformation of the initial system into yours b) what is exactly the set of solutions, being understood that it depends (at least) on an arbitrary constant : why/how that ?
– Jean Marie
yesterday
|
show 3 more comments
Not really complete, but an interesting result using simple algebraic manipulations.
Write:
$$begin{align}
a^2+b^2&=(a+b)^2-2ab\
a^3+b^3&=(a+b)(a^2+b^2-ab)=(a+b)((a+b)^2-3ab)\
a^4+b^4&=((a^2)^2+(b^2)^2)=cdots=(a+b)^4+2a^2b^2-4ab(a+b)^2\
vdots
end{align}$$
From $(2)$, we have:
$$begin{align}
(a^2+b^2)(c^2+d^2)&=((a+b)^2-2ab)((c+d)^2-2cd)=9\
&=color{red}{(a+b)^2(c+d)^2}-2ab(c+d)^2-2cd(a+b)^2-4abcd=9
end{align}$$
But from $(1)$, we know that $(a+b)(c+d)=1$, then the text in $color{red}{text{red}}$ is also equal to $1$, so the result above becomes:
$$2abcd-ab(c+d)^2-cd(a+b)^2=4tag{1*}$$
From $(3)$, we have:
$$begin{align}
(a^3+b^3)(c^3+d^3)&=color{red}{(a+b)}((a+b)^2-3ab)color{red}{(c+d)}((c+d)^2-3cd)=7\
&=((a+b)^2-3ab)((c+d)^2-3cd)=7\
&qquadvdots\
&=3abcd-ab(c+d)^2-cd(a+b)^2=2tag{2*}
end{align}$$
Adding $(1*)$ and $(2*)$, we get $abcd=-2$ and that $color{pink}{ab(c+d)^2+cd(a+b)^2=-8}$.
Now $(4)$ is really tricky, but you can write it as:
$$begin{align}
left((a+b)^4+2a^2b^2-4ab(a+b)^2right)left((c+d)^4+2c^2d^2-4cd(c+d)^2right)&=25
end{align}$$
Expanding, and we can eliminate $(a+b)^4(c+d)^4$ since it is equal to $1$. Then we have:
$$begin{align}
a^2b^2(c+d)^4-2ab(c+d)^2+c^2d^2(a+b)^4+2(abcd)^2-4abc^2d^2(a+b)^2-2cd(a+b)^2-4a^2b^2cd(c+d)^2-4abcd=12
end{align}$$
Using the fact that $abcd=-2$, then we can shorten the equation above into:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}+5ab(c+d)^2-10cd(a+b)^2+16=12tag{3*}$$
However, you can see that the text in $color{red}{text{red}}$ looks very close to the square of two sums:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}=(ab(c+d)^2+cd(a+b)^2)^2-2color{blue}{abcd(a+b)^2(c+d)^2}$$
However we already know the value of the part in $color{blue}{text{blue}}$ to be $-2
cdot 1$.
Now we can write $(3*)$ as:
$$(ab(c+d)^2+cd(a+b)^2)^2+6ab(c+d)^2-10cd(a+b)^2=-8$$
From here, you can substitute $x=ab(c+d)^2$ and $y=cd(a+b)^2$, which gives two systems of equation:
$$(x+y)^2+6x-10y=-8\
x+y=-8$$
This has one solution:
$$x=-frac{19}2,,y=frac32$$
You can try working from here.
You and I have been working on the same tracks : obtaining values for expressions involving $a+b$ and $ab$ (such as my $alpha$ and $beta$) that provide equations. Afterwards, I would say that it's a matter of luck to find the right equations...
– Jean Marie
14 hours ago
add a comment |
We are going to show that $(a,b)$ belongs to one of the two lines with equations $b=sqrt{a}$ and $b=frac{1}{sqrt{a}}$ as displayed on the following figure. It will give the answer, due to the symmetry of the system of equations with respect to the group of variables $(a,b)$ vs. $(c,d)$. Moreover, we will establish (see (*) at the bottom) that the last equation is superfluous.
Here is the explanation :
Let :
$$S_1:=a+b, S_2:=c+d, P_1:=ab, P_2:=cd$$
The system constituted by the first three equations can be written, with these variables, using classical transformations :
$$begin{cases}
(A) &S_1S_2&=&1& &\
(B) &(S_1^2-2P_1)(S_2^2-2P_2)&=&9 & implies & (C) 1-2(P_1S_2^2+P_2S_1^2)+4(P_1P_2)=9\
(D) &(S_1^3-2P_1S_1)(S_2^3-2P_2S_2)&=&7 & implies & (E) 1-3S_1S_2(P_1S_2^2+P_2S_1^2)+9(P_1P_2)=7.
end{cases}$$
(equations (C) and (E) are obtained by expansion of (B) and (D) resp., using relationship (A)).
Setting
$$alpha := P_1P_2 text{and} beta := P_1S_2^2+P_2S_1^2,$$
equations (C) and (E) become :
$$begin{cases}
(C) & 2alpha-beta&=&4\
(E) & 3alpha-beta&=&2
end{cases} implies alpha=-2 text{and} beta=-8.$$
Using the fact that $S_1S_2=1$ and $alpha=P_1P_2=-2$, equation $beta=-8$ becomes :
$$P_1 frac{1}{S_1^2} - frac{2}{P_1}S_1^2 = -8$$
i.e.,
$$(F) P_1^2 + 8 P_1S_1^2 - 2 S_1^4 =0,$$
which can be considered as a quadratic equation in variable $P_1$ giving two solutions. Due to classical condition
$$(a+b)^2 geq 2ab iff S_1^2 geq 2P_1,$$
only one of these solutions is eligible :
$$P_1=(-4+3sqrt{2})S_1^2 iff ab=(-4+3sqrt{2})(a+b)^2 iff (b-sqrt{2}a)(b-frac{sqrt{2}}{2}a)=0$$
whence the result corresponding to the figure.
The parametric equations of the two lines are
$$(a,b)=(p,p sqrt{2}) text{and} (a,b)=(p,p frac{sqrt{2}}{2}), text{for any} p neq 0$$
Due to the symmetry of equations, we have as well, for any $q neq 0$ :
$$(c,d)=(q,q sqrt{2}) text{and} (c,d)=(q,q frac{sqrt{2}}{2}).$$
A quick glance at any of the four equations show that necessarily $q=frac{1}{p}$. We find back in this way all the solutions given by @Claude Leibovici and @A. Pongrácz .
(*) In fact, the fourth equation is a consequence of the first three. Here is why :
First of all, relationship (F) is equivalent to :
$$(G) S_1^4=frac12P_1^2+4P_1S_1^2.$$
As the fourth equation can be written :
$$(H) (S_1^4+2P_1^2-4P_1S_1^2)(S_2^4+2P_2^2-4P_2S_2^2)=25,$$
using (G) in (H), we get :
$$frac52P_1^2 frac52P_2^2=25,$$
which is a tautology due to the fact that $alpha=P_1P_2=-2.$
add a comment |
This is not an answer but it is too long for a comment.
Looking at this system of equations, I had a very strange feeling (which I cannot explain).
Using a CAS, I solved equations $(1)$, $(2)$, $(3)$ for $a,b,c$ as functions of $d$ and obtained $8$ solutions which are listed below
$$left{a= frac{2}{d},b= frac{sqrt{2}}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{sqrt{2}}{d},b= frac{2}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{2}{d},b=
-frac{sqrt{2}}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{sqrt{2}}{d},b= frac{2}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{1}{d},b= -frac{sqrt{2}}{d},c= -sqrt{2} dright},left{a=
-frac{sqrt{2}}{d},b= -frac{1}{d},c= -sqrt{2} dright},left{a=
-frac{1}{d},b= frac{sqrt{2}}{d},c= sqrt{2} dright},left{a=
frac{sqrt{2}}{d},b= -frac{1}{d},c= sqrt{2} dright}$$
The problem is that, replacing in $(4)$ any of these solutions the resulting equation is $25=25$ !
4
It is not a problem, Claude: you are perfectly right : if $(a,b,c,d)$ is a solution, $(a/k, b/k , ck, dk)$ is clearly a solution as well... Besides, bonne année 2019 !
– Jean Marie
yesterday
@JeanMarie. Good to see you here ! All my best wishes. Thanks for the comment.
– Claude Leibovici
yesterday
This issue is puzzling : the fact that we have an infinite set of solutions means a kind of (algebraic) dependency of this set of equations. But, how to find it, and, in the case such a dependency is found, how can we exploit it ?
– Jean Marie
yesterday
add a comment |
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4 Answers
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4 Answers
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Hint: $ac=x, bc=y, ad=u, bd=v$, then the equations are
$x+y+u+v=1$
$x^2+y^2+u^2+v^2=9$
$x^3+y^3+u^3+v^3=7$
$x^4+y^4+u^4+v^4=25$
Use Newton-Girard to compute the elementary polynomials.
Then you have the polynomial $P(z)= (z-x)(z-y)(z-u)(z-v)$ with variable $z$.
Solve the quartic equation $P(z)=0$, and there you have the values $x,y,u,v$ in some order.
Note that not in any order: $xv=yu$ must be true, see the definition of these variables.
Of course, once you have $x,y,u,v$, it is easy to compute $a,b,c,d$.
P.S. By this way we can get:
$${x,y,u,v}={-1,2,sqrt2,-sqrt2},$$ which gives $abcd=-2.$
Up to symmetry, the solution is $(a,b,c,d)= (t, -sqrt{2}t, -frac{1}{t}, -frac{sqrt{2}}{t})$ for any $tneq 0$.
(By up to symmetry, I mean you can switch $a$ and $b$, you can switch $c$ and $d$, and you can switch the pair $(a,b)$ with $(c,d)$, so there are $8$ symmetries.)
2
I solved your system. What is the rest?
– Michael Rozenberg
yesterday
2
Yes, I see now. We can kill it.
– Michael Rozenberg
yesterday
I am sorry but I don't find your derivation very clear. Besides, a) the solution you propose doesn't work : from the four equations $(1) ac=-1=x, (2) bc=2=y, (3) ad=sqrt{2}=u, (4) bd=-sqrt{2}=v$, if I divide (1) by (3), I get $c/d=-1/sqrt{2}$, whereas, if I divide (2) by (3), I get $c/d=sqrt{2}$, which is contradictory. b) See the infinite family of solutions provided by Claude.
– Jean Marie
yesterday
1
@JeanMarie I am sorry if you don't find it clear, but could you be more specific? I checked, and it seems to me that the solution (given by @MichaelRozenberg) is correct for $x,y,u,v$. From that it should be a routine taks to solve the original system of equations. You made a mistake though: as I said, you cannot randomly pick which root is which variable. It should be $ab=-1, cd=2, ac=sqrt{2}, bd=-sqrt{2}$, or one of the other seven symmetrical choices. Keep in mind that $xv=yu$.
– A. Pongrácz
yesterday
I realize that I have misunderstood your set equality $${x,y,u,v}={-1,2,sqrt2,-sqrt2}$$ that I have interpreted as a t-uple equality $$(x,y,u,v)=(-1,2,sqrt2,-sqrt2)$$... But what I find not evident to understand is a) the (clever !) transformation of the initial system into yours b) what is exactly the set of solutions, being understood that it depends (at least) on an arbitrary constant : why/how that ?
– Jean Marie
yesterday
|
show 3 more comments
Hint: $ac=x, bc=y, ad=u, bd=v$, then the equations are
$x+y+u+v=1$
$x^2+y^2+u^2+v^2=9$
$x^3+y^3+u^3+v^3=7$
$x^4+y^4+u^4+v^4=25$
Use Newton-Girard to compute the elementary polynomials.
Then you have the polynomial $P(z)= (z-x)(z-y)(z-u)(z-v)$ with variable $z$.
Solve the quartic equation $P(z)=0$, and there you have the values $x,y,u,v$ in some order.
Note that not in any order: $xv=yu$ must be true, see the definition of these variables.
Of course, once you have $x,y,u,v$, it is easy to compute $a,b,c,d$.
P.S. By this way we can get:
$${x,y,u,v}={-1,2,sqrt2,-sqrt2},$$ which gives $abcd=-2.$
Up to symmetry, the solution is $(a,b,c,d)= (t, -sqrt{2}t, -frac{1}{t}, -frac{sqrt{2}}{t})$ for any $tneq 0$.
(By up to symmetry, I mean you can switch $a$ and $b$, you can switch $c$ and $d$, and you can switch the pair $(a,b)$ with $(c,d)$, so there are $8$ symmetries.)
2
I solved your system. What is the rest?
– Michael Rozenberg
yesterday
2
Yes, I see now. We can kill it.
– Michael Rozenberg
yesterday
I am sorry but I don't find your derivation very clear. Besides, a) the solution you propose doesn't work : from the four equations $(1) ac=-1=x, (2) bc=2=y, (3) ad=sqrt{2}=u, (4) bd=-sqrt{2}=v$, if I divide (1) by (3), I get $c/d=-1/sqrt{2}$, whereas, if I divide (2) by (3), I get $c/d=sqrt{2}$, which is contradictory. b) See the infinite family of solutions provided by Claude.
– Jean Marie
yesterday
1
@JeanMarie I am sorry if you don't find it clear, but could you be more specific? I checked, and it seems to me that the solution (given by @MichaelRozenberg) is correct for $x,y,u,v$. From that it should be a routine taks to solve the original system of equations. You made a mistake though: as I said, you cannot randomly pick which root is which variable. It should be $ab=-1, cd=2, ac=sqrt{2}, bd=-sqrt{2}$, or one of the other seven symmetrical choices. Keep in mind that $xv=yu$.
– A. Pongrácz
yesterday
I realize that I have misunderstood your set equality $${x,y,u,v}={-1,2,sqrt2,-sqrt2}$$ that I have interpreted as a t-uple equality $$(x,y,u,v)=(-1,2,sqrt2,-sqrt2)$$... But what I find not evident to understand is a) the (clever !) transformation of the initial system into yours b) what is exactly the set of solutions, being understood that it depends (at least) on an arbitrary constant : why/how that ?
– Jean Marie
yesterday
|
show 3 more comments
Hint: $ac=x, bc=y, ad=u, bd=v$, then the equations are
$x+y+u+v=1$
$x^2+y^2+u^2+v^2=9$
$x^3+y^3+u^3+v^3=7$
$x^4+y^4+u^4+v^4=25$
Use Newton-Girard to compute the elementary polynomials.
Then you have the polynomial $P(z)= (z-x)(z-y)(z-u)(z-v)$ with variable $z$.
Solve the quartic equation $P(z)=0$, and there you have the values $x,y,u,v$ in some order.
Note that not in any order: $xv=yu$ must be true, see the definition of these variables.
Of course, once you have $x,y,u,v$, it is easy to compute $a,b,c,d$.
P.S. By this way we can get:
$${x,y,u,v}={-1,2,sqrt2,-sqrt2},$$ which gives $abcd=-2.$
Up to symmetry, the solution is $(a,b,c,d)= (t, -sqrt{2}t, -frac{1}{t}, -frac{sqrt{2}}{t})$ for any $tneq 0$.
(By up to symmetry, I mean you can switch $a$ and $b$, you can switch $c$ and $d$, and you can switch the pair $(a,b)$ with $(c,d)$, so there are $8$ symmetries.)
Hint: $ac=x, bc=y, ad=u, bd=v$, then the equations are
$x+y+u+v=1$
$x^2+y^2+u^2+v^2=9$
$x^3+y^3+u^3+v^3=7$
$x^4+y^4+u^4+v^4=25$
Use Newton-Girard to compute the elementary polynomials.
Then you have the polynomial $P(z)= (z-x)(z-y)(z-u)(z-v)$ with variable $z$.
Solve the quartic equation $P(z)=0$, and there you have the values $x,y,u,v$ in some order.
Note that not in any order: $xv=yu$ must be true, see the definition of these variables.
Of course, once you have $x,y,u,v$, it is easy to compute $a,b,c,d$.
P.S. By this way we can get:
$${x,y,u,v}={-1,2,sqrt2,-sqrt2},$$ which gives $abcd=-2.$
Up to symmetry, the solution is $(a,b,c,d)= (t, -sqrt{2}t, -frac{1}{t}, -frac{sqrt{2}}{t})$ for any $tneq 0$.
(By up to symmetry, I mean you can switch $a$ and $b$, you can switch $c$ and $d$, and you can switch the pair $(a,b)$ with $(c,d)$, so there are $8$ symmetries.)
edited yesterday
answered yesterday
A. PongráczA. Pongrácz
5,8001929
5,8001929
2
I solved your system. What is the rest?
– Michael Rozenberg
yesterday
2
Yes, I see now. We can kill it.
– Michael Rozenberg
yesterday
I am sorry but I don't find your derivation very clear. Besides, a) the solution you propose doesn't work : from the four equations $(1) ac=-1=x, (2) bc=2=y, (3) ad=sqrt{2}=u, (4) bd=-sqrt{2}=v$, if I divide (1) by (3), I get $c/d=-1/sqrt{2}$, whereas, if I divide (2) by (3), I get $c/d=sqrt{2}$, which is contradictory. b) See the infinite family of solutions provided by Claude.
– Jean Marie
yesterday
1
@JeanMarie I am sorry if you don't find it clear, but could you be more specific? I checked, and it seems to me that the solution (given by @MichaelRozenberg) is correct for $x,y,u,v$. From that it should be a routine taks to solve the original system of equations. You made a mistake though: as I said, you cannot randomly pick which root is which variable. It should be $ab=-1, cd=2, ac=sqrt{2}, bd=-sqrt{2}$, or one of the other seven symmetrical choices. Keep in mind that $xv=yu$.
– A. Pongrácz
yesterday
I realize that I have misunderstood your set equality $${x,y,u,v}={-1,2,sqrt2,-sqrt2}$$ that I have interpreted as a t-uple equality $$(x,y,u,v)=(-1,2,sqrt2,-sqrt2)$$... But what I find not evident to understand is a) the (clever !) transformation of the initial system into yours b) what is exactly the set of solutions, being understood that it depends (at least) on an arbitrary constant : why/how that ?
– Jean Marie
yesterday
|
show 3 more comments
2
I solved your system. What is the rest?
– Michael Rozenberg
yesterday
2
Yes, I see now. We can kill it.
– Michael Rozenberg
yesterday
I am sorry but I don't find your derivation very clear. Besides, a) the solution you propose doesn't work : from the four equations $(1) ac=-1=x, (2) bc=2=y, (3) ad=sqrt{2}=u, (4) bd=-sqrt{2}=v$, if I divide (1) by (3), I get $c/d=-1/sqrt{2}$, whereas, if I divide (2) by (3), I get $c/d=sqrt{2}$, which is contradictory. b) See the infinite family of solutions provided by Claude.
– Jean Marie
yesterday
1
@JeanMarie I am sorry if you don't find it clear, but could you be more specific? I checked, and it seems to me that the solution (given by @MichaelRozenberg) is correct for $x,y,u,v$. From that it should be a routine taks to solve the original system of equations. You made a mistake though: as I said, you cannot randomly pick which root is which variable. It should be $ab=-1, cd=2, ac=sqrt{2}, bd=-sqrt{2}$, or one of the other seven symmetrical choices. Keep in mind that $xv=yu$.
– A. Pongrácz
yesterday
I realize that I have misunderstood your set equality $${x,y,u,v}={-1,2,sqrt2,-sqrt2}$$ that I have interpreted as a t-uple equality $$(x,y,u,v)=(-1,2,sqrt2,-sqrt2)$$... But what I find not evident to understand is a) the (clever !) transformation of the initial system into yours b) what is exactly the set of solutions, being understood that it depends (at least) on an arbitrary constant : why/how that ?
– Jean Marie
yesterday
2
2
I solved your system. What is the rest?
– Michael Rozenberg
yesterday
I solved your system. What is the rest?
– Michael Rozenberg
yesterday
2
2
Yes, I see now. We can kill it.
– Michael Rozenberg
yesterday
Yes, I see now. We can kill it.
– Michael Rozenberg
yesterday
I am sorry but I don't find your derivation very clear. Besides, a) the solution you propose doesn't work : from the four equations $(1) ac=-1=x, (2) bc=2=y, (3) ad=sqrt{2}=u, (4) bd=-sqrt{2}=v$, if I divide (1) by (3), I get $c/d=-1/sqrt{2}$, whereas, if I divide (2) by (3), I get $c/d=sqrt{2}$, which is contradictory. b) See the infinite family of solutions provided by Claude.
– Jean Marie
yesterday
I am sorry but I don't find your derivation very clear. Besides, a) the solution you propose doesn't work : from the four equations $(1) ac=-1=x, (2) bc=2=y, (3) ad=sqrt{2}=u, (4) bd=-sqrt{2}=v$, if I divide (1) by (3), I get $c/d=-1/sqrt{2}$, whereas, if I divide (2) by (3), I get $c/d=sqrt{2}$, which is contradictory. b) See the infinite family of solutions provided by Claude.
– Jean Marie
yesterday
1
1
@JeanMarie I am sorry if you don't find it clear, but could you be more specific? I checked, and it seems to me that the solution (given by @MichaelRozenberg) is correct for $x,y,u,v$. From that it should be a routine taks to solve the original system of equations. You made a mistake though: as I said, you cannot randomly pick which root is which variable. It should be $ab=-1, cd=2, ac=sqrt{2}, bd=-sqrt{2}$, or one of the other seven symmetrical choices. Keep in mind that $xv=yu$.
– A. Pongrácz
yesterday
@JeanMarie I am sorry if you don't find it clear, but could you be more specific? I checked, and it seems to me that the solution (given by @MichaelRozenberg) is correct for $x,y,u,v$. From that it should be a routine taks to solve the original system of equations. You made a mistake though: as I said, you cannot randomly pick which root is which variable. It should be $ab=-1, cd=2, ac=sqrt{2}, bd=-sqrt{2}$, or one of the other seven symmetrical choices. Keep in mind that $xv=yu$.
– A. Pongrácz
yesterday
I realize that I have misunderstood your set equality $${x,y,u,v}={-1,2,sqrt2,-sqrt2}$$ that I have interpreted as a t-uple equality $$(x,y,u,v)=(-1,2,sqrt2,-sqrt2)$$... But what I find not evident to understand is a) the (clever !) transformation of the initial system into yours b) what is exactly the set of solutions, being understood that it depends (at least) on an arbitrary constant : why/how that ?
– Jean Marie
yesterday
I realize that I have misunderstood your set equality $${x,y,u,v}={-1,2,sqrt2,-sqrt2}$$ that I have interpreted as a t-uple equality $$(x,y,u,v)=(-1,2,sqrt2,-sqrt2)$$... But what I find not evident to understand is a) the (clever !) transformation of the initial system into yours b) what is exactly the set of solutions, being understood that it depends (at least) on an arbitrary constant : why/how that ?
– Jean Marie
yesterday
|
show 3 more comments
Not really complete, but an interesting result using simple algebraic manipulations.
Write:
$$begin{align}
a^2+b^2&=(a+b)^2-2ab\
a^3+b^3&=(a+b)(a^2+b^2-ab)=(a+b)((a+b)^2-3ab)\
a^4+b^4&=((a^2)^2+(b^2)^2)=cdots=(a+b)^4+2a^2b^2-4ab(a+b)^2\
vdots
end{align}$$
From $(2)$, we have:
$$begin{align}
(a^2+b^2)(c^2+d^2)&=((a+b)^2-2ab)((c+d)^2-2cd)=9\
&=color{red}{(a+b)^2(c+d)^2}-2ab(c+d)^2-2cd(a+b)^2-4abcd=9
end{align}$$
But from $(1)$, we know that $(a+b)(c+d)=1$, then the text in $color{red}{text{red}}$ is also equal to $1$, so the result above becomes:
$$2abcd-ab(c+d)^2-cd(a+b)^2=4tag{1*}$$
From $(3)$, we have:
$$begin{align}
(a^3+b^3)(c^3+d^3)&=color{red}{(a+b)}((a+b)^2-3ab)color{red}{(c+d)}((c+d)^2-3cd)=7\
&=((a+b)^2-3ab)((c+d)^2-3cd)=7\
&qquadvdots\
&=3abcd-ab(c+d)^2-cd(a+b)^2=2tag{2*}
end{align}$$
Adding $(1*)$ and $(2*)$, we get $abcd=-2$ and that $color{pink}{ab(c+d)^2+cd(a+b)^2=-8}$.
Now $(4)$ is really tricky, but you can write it as:
$$begin{align}
left((a+b)^4+2a^2b^2-4ab(a+b)^2right)left((c+d)^4+2c^2d^2-4cd(c+d)^2right)&=25
end{align}$$
Expanding, and we can eliminate $(a+b)^4(c+d)^4$ since it is equal to $1$. Then we have:
$$begin{align}
a^2b^2(c+d)^4-2ab(c+d)^2+c^2d^2(a+b)^4+2(abcd)^2-4abc^2d^2(a+b)^2-2cd(a+b)^2-4a^2b^2cd(c+d)^2-4abcd=12
end{align}$$
Using the fact that $abcd=-2$, then we can shorten the equation above into:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}+5ab(c+d)^2-10cd(a+b)^2+16=12tag{3*}$$
However, you can see that the text in $color{red}{text{red}}$ looks very close to the square of two sums:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}=(ab(c+d)^2+cd(a+b)^2)^2-2color{blue}{abcd(a+b)^2(c+d)^2}$$
However we already know the value of the part in $color{blue}{text{blue}}$ to be $-2
cdot 1$.
Now we can write $(3*)$ as:
$$(ab(c+d)^2+cd(a+b)^2)^2+6ab(c+d)^2-10cd(a+b)^2=-8$$
From here, you can substitute $x=ab(c+d)^2$ and $y=cd(a+b)^2$, which gives two systems of equation:
$$(x+y)^2+6x-10y=-8\
x+y=-8$$
This has one solution:
$$x=-frac{19}2,,y=frac32$$
You can try working from here.
You and I have been working on the same tracks : obtaining values for expressions involving $a+b$ and $ab$ (such as my $alpha$ and $beta$) that provide equations. Afterwards, I would say that it's a matter of luck to find the right equations...
– Jean Marie
14 hours ago
add a comment |
Not really complete, but an interesting result using simple algebraic manipulations.
Write:
$$begin{align}
a^2+b^2&=(a+b)^2-2ab\
a^3+b^3&=(a+b)(a^2+b^2-ab)=(a+b)((a+b)^2-3ab)\
a^4+b^4&=((a^2)^2+(b^2)^2)=cdots=(a+b)^4+2a^2b^2-4ab(a+b)^2\
vdots
end{align}$$
From $(2)$, we have:
$$begin{align}
(a^2+b^2)(c^2+d^2)&=((a+b)^2-2ab)((c+d)^2-2cd)=9\
&=color{red}{(a+b)^2(c+d)^2}-2ab(c+d)^2-2cd(a+b)^2-4abcd=9
end{align}$$
But from $(1)$, we know that $(a+b)(c+d)=1$, then the text in $color{red}{text{red}}$ is also equal to $1$, so the result above becomes:
$$2abcd-ab(c+d)^2-cd(a+b)^2=4tag{1*}$$
From $(3)$, we have:
$$begin{align}
(a^3+b^3)(c^3+d^3)&=color{red}{(a+b)}((a+b)^2-3ab)color{red}{(c+d)}((c+d)^2-3cd)=7\
&=((a+b)^2-3ab)((c+d)^2-3cd)=7\
&qquadvdots\
&=3abcd-ab(c+d)^2-cd(a+b)^2=2tag{2*}
end{align}$$
Adding $(1*)$ and $(2*)$, we get $abcd=-2$ and that $color{pink}{ab(c+d)^2+cd(a+b)^2=-8}$.
Now $(4)$ is really tricky, but you can write it as:
$$begin{align}
left((a+b)^4+2a^2b^2-4ab(a+b)^2right)left((c+d)^4+2c^2d^2-4cd(c+d)^2right)&=25
end{align}$$
Expanding, and we can eliminate $(a+b)^4(c+d)^4$ since it is equal to $1$. Then we have:
$$begin{align}
a^2b^2(c+d)^4-2ab(c+d)^2+c^2d^2(a+b)^4+2(abcd)^2-4abc^2d^2(a+b)^2-2cd(a+b)^2-4a^2b^2cd(c+d)^2-4abcd=12
end{align}$$
Using the fact that $abcd=-2$, then we can shorten the equation above into:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}+5ab(c+d)^2-10cd(a+b)^2+16=12tag{3*}$$
However, you can see that the text in $color{red}{text{red}}$ looks very close to the square of two sums:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}=(ab(c+d)^2+cd(a+b)^2)^2-2color{blue}{abcd(a+b)^2(c+d)^2}$$
However we already know the value of the part in $color{blue}{text{blue}}$ to be $-2
cdot 1$.
Now we can write $(3*)$ as:
$$(ab(c+d)^2+cd(a+b)^2)^2+6ab(c+d)^2-10cd(a+b)^2=-8$$
From here, you can substitute $x=ab(c+d)^2$ and $y=cd(a+b)^2$, which gives two systems of equation:
$$(x+y)^2+6x-10y=-8\
x+y=-8$$
This has one solution:
$$x=-frac{19}2,,y=frac32$$
You can try working from here.
You and I have been working on the same tracks : obtaining values for expressions involving $a+b$ and $ab$ (such as my $alpha$ and $beta$) that provide equations. Afterwards, I would say that it's a matter of luck to find the right equations...
– Jean Marie
14 hours ago
add a comment |
Not really complete, but an interesting result using simple algebraic manipulations.
Write:
$$begin{align}
a^2+b^2&=(a+b)^2-2ab\
a^3+b^3&=(a+b)(a^2+b^2-ab)=(a+b)((a+b)^2-3ab)\
a^4+b^4&=((a^2)^2+(b^2)^2)=cdots=(a+b)^4+2a^2b^2-4ab(a+b)^2\
vdots
end{align}$$
From $(2)$, we have:
$$begin{align}
(a^2+b^2)(c^2+d^2)&=((a+b)^2-2ab)((c+d)^2-2cd)=9\
&=color{red}{(a+b)^2(c+d)^2}-2ab(c+d)^2-2cd(a+b)^2-4abcd=9
end{align}$$
But from $(1)$, we know that $(a+b)(c+d)=1$, then the text in $color{red}{text{red}}$ is also equal to $1$, so the result above becomes:
$$2abcd-ab(c+d)^2-cd(a+b)^2=4tag{1*}$$
From $(3)$, we have:
$$begin{align}
(a^3+b^3)(c^3+d^3)&=color{red}{(a+b)}((a+b)^2-3ab)color{red}{(c+d)}((c+d)^2-3cd)=7\
&=((a+b)^2-3ab)((c+d)^2-3cd)=7\
&qquadvdots\
&=3abcd-ab(c+d)^2-cd(a+b)^2=2tag{2*}
end{align}$$
Adding $(1*)$ and $(2*)$, we get $abcd=-2$ and that $color{pink}{ab(c+d)^2+cd(a+b)^2=-8}$.
Now $(4)$ is really tricky, but you can write it as:
$$begin{align}
left((a+b)^4+2a^2b^2-4ab(a+b)^2right)left((c+d)^4+2c^2d^2-4cd(c+d)^2right)&=25
end{align}$$
Expanding, and we can eliminate $(a+b)^4(c+d)^4$ since it is equal to $1$. Then we have:
$$begin{align}
a^2b^2(c+d)^4-2ab(c+d)^2+c^2d^2(a+b)^4+2(abcd)^2-4abc^2d^2(a+b)^2-2cd(a+b)^2-4a^2b^2cd(c+d)^2-4abcd=12
end{align}$$
Using the fact that $abcd=-2$, then we can shorten the equation above into:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}+5ab(c+d)^2-10cd(a+b)^2+16=12tag{3*}$$
However, you can see that the text in $color{red}{text{red}}$ looks very close to the square of two sums:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}=(ab(c+d)^2+cd(a+b)^2)^2-2color{blue}{abcd(a+b)^2(c+d)^2}$$
However we already know the value of the part in $color{blue}{text{blue}}$ to be $-2
cdot 1$.
Now we can write $(3*)$ as:
$$(ab(c+d)^2+cd(a+b)^2)^2+6ab(c+d)^2-10cd(a+b)^2=-8$$
From here, you can substitute $x=ab(c+d)^2$ and $y=cd(a+b)^2$, which gives two systems of equation:
$$(x+y)^2+6x-10y=-8\
x+y=-8$$
This has one solution:
$$x=-frac{19}2,,y=frac32$$
You can try working from here.
Not really complete, but an interesting result using simple algebraic manipulations.
Write:
$$begin{align}
a^2+b^2&=(a+b)^2-2ab\
a^3+b^3&=(a+b)(a^2+b^2-ab)=(a+b)((a+b)^2-3ab)\
a^4+b^4&=((a^2)^2+(b^2)^2)=cdots=(a+b)^4+2a^2b^2-4ab(a+b)^2\
vdots
end{align}$$
From $(2)$, we have:
$$begin{align}
(a^2+b^2)(c^2+d^2)&=((a+b)^2-2ab)((c+d)^2-2cd)=9\
&=color{red}{(a+b)^2(c+d)^2}-2ab(c+d)^2-2cd(a+b)^2-4abcd=9
end{align}$$
But from $(1)$, we know that $(a+b)(c+d)=1$, then the text in $color{red}{text{red}}$ is also equal to $1$, so the result above becomes:
$$2abcd-ab(c+d)^2-cd(a+b)^2=4tag{1*}$$
From $(3)$, we have:
$$begin{align}
(a^3+b^3)(c^3+d^3)&=color{red}{(a+b)}((a+b)^2-3ab)color{red}{(c+d)}((c+d)^2-3cd)=7\
&=((a+b)^2-3ab)((c+d)^2-3cd)=7\
&qquadvdots\
&=3abcd-ab(c+d)^2-cd(a+b)^2=2tag{2*}
end{align}$$
Adding $(1*)$ and $(2*)$, we get $abcd=-2$ and that $color{pink}{ab(c+d)^2+cd(a+b)^2=-8}$.
Now $(4)$ is really tricky, but you can write it as:
$$begin{align}
left((a+b)^4+2a^2b^2-4ab(a+b)^2right)left((c+d)^4+2c^2d^2-4cd(c+d)^2right)&=25
end{align}$$
Expanding, and we can eliminate $(a+b)^4(c+d)^4$ since it is equal to $1$. Then we have:
$$begin{align}
a^2b^2(c+d)^4-2ab(c+d)^2+c^2d^2(a+b)^4+2(abcd)^2-4abc^2d^2(a+b)^2-2cd(a+b)^2-4a^2b^2cd(c+d)^2-4abcd=12
end{align}$$
Using the fact that $abcd=-2$, then we can shorten the equation above into:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}+5ab(c+d)^2-10cd(a+b)^2+16=12tag{3*}$$
However, you can see that the text in $color{red}{text{red}}$ looks very close to the square of two sums:
$$color{red}{a^2b^2(c+d)^4+c^2d^2(a+b)^4}=(ab(c+d)^2+cd(a+b)^2)^2-2color{blue}{abcd(a+b)^2(c+d)^2}$$
However we already know the value of the part in $color{blue}{text{blue}}$ to be $-2
cdot 1$.
Now we can write $(3*)$ as:
$$(ab(c+d)^2+cd(a+b)^2)^2+6ab(c+d)^2-10cd(a+b)^2=-8$$
From here, you can substitute $x=ab(c+d)^2$ and $y=cd(a+b)^2$, which gives two systems of equation:
$$(x+y)^2+6x-10y=-8\
x+y=-8$$
This has one solution:
$$x=-frac{19}2,,y=frac32$$
You can try working from here.
edited yesterday
answered yesterday
John GlennJohn Glenn
1,957424
1,957424
You and I have been working on the same tracks : obtaining values for expressions involving $a+b$ and $ab$ (such as my $alpha$ and $beta$) that provide equations. Afterwards, I would say that it's a matter of luck to find the right equations...
– Jean Marie
14 hours ago
add a comment |
You and I have been working on the same tracks : obtaining values for expressions involving $a+b$ and $ab$ (such as my $alpha$ and $beta$) that provide equations. Afterwards, I would say that it's a matter of luck to find the right equations...
– Jean Marie
14 hours ago
You and I have been working on the same tracks : obtaining values for expressions involving $a+b$ and $ab$ (such as my $alpha$ and $beta$) that provide equations. Afterwards, I would say that it's a matter of luck to find the right equations...
– Jean Marie
14 hours ago
You and I have been working on the same tracks : obtaining values for expressions involving $a+b$ and $ab$ (such as my $alpha$ and $beta$) that provide equations. Afterwards, I would say that it's a matter of luck to find the right equations...
– Jean Marie
14 hours ago
add a comment |
We are going to show that $(a,b)$ belongs to one of the two lines with equations $b=sqrt{a}$ and $b=frac{1}{sqrt{a}}$ as displayed on the following figure. It will give the answer, due to the symmetry of the system of equations with respect to the group of variables $(a,b)$ vs. $(c,d)$. Moreover, we will establish (see (*) at the bottom) that the last equation is superfluous.
Here is the explanation :
Let :
$$S_1:=a+b, S_2:=c+d, P_1:=ab, P_2:=cd$$
The system constituted by the first three equations can be written, with these variables, using classical transformations :
$$begin{cases}
(A) &S_1S_2&=&1& &\
(B) &(S_1^2-2P_1)(S_2^2-2P_2)&=&9 & implies & (C) 1-2(P_1S_2^2+P_2S_1^2)+4(P_1P_2)=9\
(D) &(S_1^3-2P_1S_1)(S_2^3-2P_2S_2)&=&7 & implies & (E) 1-3S_1S_2(P_1S_2^2+P_2S_1^2)+9(P_1P_2)=7.
end{cases}$$
(equations (C) and (E) are obtained by expansion of (B) and (D) resp., using relationship (A)).
Setting
$$alpha := P_1P_2 text{and} beta := P_1S_2^2+P_2S_1^2,$$
equations (C) and (E) become :
$$begin{cases}
(C) & 2alpha-beta&=&4\
(E) & 3alpha-beta&=&2
end{cases} implies alpha=-2 text{and} beta=-8.$$
Using the fact that $S_1S_2=1$ and $alpha=P_1P_2=-2$, equation $beta=-8$ becomes :
$$P_1 frac{1}{S_1^2} - frac{2}{P_1}S_1^2 = -8$$
i.e.,
$$(F) P_1^2 + 8 P_1S_1^2 - 2 S_1^4 =0,$$
which can be considered as a quadratic equation in variable $P_1$ giving two solutions. Due to classical condition
$$(a+b)^2 geq 2ab iff S_1^2 geq 2P_1,$$
only one of these solutions is eligible :
$$P_1=(-4+3sqrt{2})S_1^2 iff ab=(-4+3sqrt{2})(a+b)^2 iff (b-sqrt{2}a)(b-frac{sqrt{2}}{2}a)=0$$
whence the result corresponding to the figure.
The parametric equations of the two lines are
$$(a,b)=(p,p sqrt{2}) text{and} (a,b)=(p,p frac{sqrt{2}}{2}), text{for any} p neq 0$$
Due to the symmetry of equations, we have as well, for any $q neq 0$ :
$$(c,d)=(q,q sqrt{2}) text{and} (c,d)=(q,q frac{sqrt{2}}{2}).$$
A quick glance at any of the four equations show that necessarily $q=frac{1}{p}$. We find back in this way all the solutions given by @Claude Leibovici and @A. Pongrácz .
(*) In fact, the fourth equation is a consequence of the first three. Here is why :
First of all, relationship (F) is equivalent to :
$$(G) S_1^4=frac12P_1^2+4P_1S_1^2.$$
As the fourth equation can be written :
$$(H) (S_1^4+2P_1^2-4P_1S_1^2)(S_2^4+2P_2^2-4P_2S_2^2)=25,$$
using (G) in (H), we get :
$$frac52P_1^2 frac52P_2^2=25,$$
which is a tautology due to the fact that $alpha=P_1P_2=-2.$
add a comment |
We are going to show that $(a,b)$ belongs to one of the two lines with equations $b=sqrt{a}$ and $b=frac{1}{sqrt{a}}$ as displayed on the following figure. It will give the answer, due to the symmetry of the system of equations with respect to the group of variables $(a,b)$ vs. $(c,d)$. Moreover, we will establish (see (*) at the bottom) that the last equation is superfluous.
Here is the explanation :
Let :
$$S_1:=a+b, S_2:=c+d, P_1:=ab, P_2:=cd$$
The system constituted by the first three equations can be written, with these variables, using classical transformations :
$$begin{cases}
(A) &S_1S_2&=&1& &\
(B) &(S_1^2-2P_1)(S_2^2-2P_2)&=&9 & implies & (C) 1-2(P_1S_2^2+P_2S_1^2)+4(P_1P_2)=9\
(D) &(S_1^3-2P_1S_1)(S_2^3-2P_2S_2)&=&7 & implies & (E) 1-3S_1S_2(P_1S_2^2+P_2S_1^2)+9(P_1P_2)=7.
end{cases}$$
(equations (C) and (E) are obtained by expansion of (B) and (D) resp., using relationship (A)).
Setting
$$alpha := P_1P_2 text{and} beta := P_1S_2^2+P_2S_1^2,$$
equations (C) and (E) become :
$$begin{cases}
(C) & 2alpha-beta&=&4\
(E) & 3alpha-beta&=&2
end{cases} implies alpha=-2 text{and} beta=-8.$$
Using the fact that $S_1S_2=1$ and $alpha=P_1P_2=-2$, equation $beta=-8$ becomes :
$$P_1 frac{1}{S_1^2} - frac{2}{P_1}S_1^2 = -8$$
i.e.,
$$(F) P_1^2 + 8 P_1S_1^2 - 2 S_1^4 =0,$$
which can be considered as a quadratic equation in variable $P_1$ giving two solutions. Due to classical condition
$$(a+b)^2 geq 2ab iff S_1^2 geq 2P_1,$$
only one of these solutions is eligible :
$$P_1=(-4+3sqrt{2})S_1^2 iff ab=(-4+3sqrt{2})(a+b)^2 iff (b-sqrt{2}a)(b-frac{sqrt{2}}{2}a)=0$$
whence the result corresponding to the figure.
The parametric equations of the two lines are
$$(a,b)=(p,p sqrt{2}) text{and} (a,b)=(p,p frac{sqrt{2}}{2}), text{for any} p neq 0$$
Due to the symmetry of equations, we have as well, for any $q neq 0$ :
$$(c,d)=(q,q sqrt{2}) text{and} (c,d)=(q,q frac{sqrt{2}}{2}).$$
A quick glance at any of the four equations show that necessarily $q=frac{1}{p}$. We find back in this way all the solutions given by @Claude Leibovici and @A. Pongrácz .
(*) In fact, the fourth equation is a consequence of the first three. Here is why :
First of all, relationship (F) is equivalent to :
$$(G) S_1^4=frac12P_1^2+4P_1S_1^2.$$
As the fourth equation can be written :
$$(H) (S_1^4+2P_1^2-4P_1S_1^2)(S_2^4+2P_2^2-4P_2S_2^2)=25,$$
using (G) in (H), we get :
$$frac52P_1^2 frac52P_2^2=25,$$
which is a tautology due to the fact that $alpha=P_1P_2=-2.$
add a comment |
We are going to show that $(a,b)$ belongs to one of the two lines with equations $b=sqrt{a}$ and $b=frac{1}{sqrt{a}}$ as displayed on the following figure. It will give the answer, due to the symmetry of the system of equations with respect to the group of variables $(a,b)$ vs. $(c,d)$. Moreover, we will establish (see (*) at the bottom) that the last equation is superfluous.
Here is the explanation :
Let :
$$S_1:=a+b, S_2:=c+d, P_1:=ab, P_2:=cd$$
The system constituted by the first three equations can be written, with these variables, using classical transformations :
$$begin{cases}
(A) &S_1S_2&=&1& &\
(B) &(S_1^2-2P_1)(S_2^2-2P_2)&=&9 & implies & (C) 1-2(P_1S_2^2+P_2S_1^2)+4(P_1P_2)=9\
(D) &(S_1^3-2P_1S_1)(S_2^3-2P_2S_2)&=&7 & implies & (E) 1-3S_1S_2(P_1S_2^2+P_2S_1^2)+9(P_1P_2)=7.
end{cases}$$
(equations (C) and (E) are obtained by expansion of (B) and (D) resp., using relationship (A)).
Setting
$$alpha := P_1P_2 text{and} beta := P_1S_2^2+P_2S_1^2,$$
equations (C) and (E) become :
$$begin{cases}
(C) & 2alpha-beta&=&4\
(E) & 3alpha-beta&=&2
end{cases} implies alpha=-2 text{and} beta=-8.$$
Using the fact that $S_1S_2=1$ and $alpha=P_1P_2=-2$, equation $beta=-8$ becomes :
$$P_1 frac{1}{S_1^2} - frac{2}{P_1}S_1^2 = -8$$
i.e.,
$$(F) P_1^2 + 8 P_1S_1^2 - 2 S_1^4 =0,$$
which can be considered as a quadratic equation in variable $P_1$ giving two solutions. Due to classical condition
$$(a+b)^2 geq 2ab iff S_1^2 geq 2P_1,$$
only one of these solutions is eligible :
$$P_1=(-4+3sqrt{2})S_1^2 iff ab=(-4+3sqrt{2})(a+b)^2 iff (b-sqrt{2}a)(b-frac{sqrt{2}}{2}a)=0$$
whence the result corresponding to the figure.
The parametric equations of the two lines are
$$(a,b)=(p,p sqrt{2}) text{and} (a,b)=(p,p frac{sqrt{2}}{2}), text{for any} p neq 0$$
Due to the symmetry of equations, we have as well, for any $q neq 0$ :
$$(c,d)=(q,q sqrt{2}) text{and} (c,d)=(q,q frac{sqrt{2}}{2}).$$
A quick glance at any of the four equations show that necessarily $q=frac{1}{p}$. We find back in this way all the solutions given by @Claude Leibovici and @A. Pongrácz .
(*) In fact, the fourth equation is a consequence of the first three. Here is why :
First of all, relationship (F) is equivalent to :
$$(G) S_1^4=frac12P_1^2+4P_1S_1^2.$$
As the fourth equation can be written :
$$(H) (S_1^4+2P_1^2-4P_1S_1^2)(S_2^4+2P_2^2-4P_2S_2^2)=25,$$
using (G) in (H), we get :
$$frac52P_1^2 frac52P_2^2=25,$$
which is a tautology due to the fact that $alpha=P_1P_2=-2.$
We are going to show that $(a,b)$ belongs to one of the two lines with equations $b=sqrt{a}$ and $b=frac{1}{sqrt{a}}$ as displayed on the following figure. It will give the answer, due to the symmetry of the system of equations with respect to the group of variables $(a,b)$ vs. $(c,d)$. Moreover, we will establish (see (*) at the bottom) that the last equation is superfluous.
Here is the explanation :
Let :
$$S_1:=a+b, S_2:=c+d, P_1:=ab, P_2:=cd$$
The system constituted by the first three equations can be written, with these variables, using classical transformations :
$$begin{cases}
(A) &S_1S_2&=&1& &\
(B) &(S_1^2-2P_1)(S_2^2-2P_2)&=&9 & implies & (C) 1-2(P_1S_2^2+P_2S_1^2)+4(P_1P_2)=9\
(D) &(S_1^3-2P_1S_1)(S_2^3-2P_2S_2)&=&7 & implies & (E) 1-3S_1S_2(P_1S_2^2+P_2S_1^2)+9(P_1P_2)=7.
end{cases}$$
(equations (C) and (E) are obtained by expansion of (B) and (D) resp., using relationship (A)).
Setting
$$alpha := P_1P_2 text{and} beta := P_1S_2^2+P_2S_1^2,$$
equations (C) and (E) become :
$$begin{cases}
(C) & 2alpha-beta&=&4\
(E) & 3alpha-beta&=&2
end{cases} implies alpha=-2 text{and} beta=-8.$$
Using the fact that $S_1S_2=1$ and $alpha=P_1P_2=-2$, equation $beta=-8$ becomes :
$$P_1 frac{1}{S_1^2} - frac{2}{P_1}S_1^2 = -8$$
i.e.,
$$(F) P_1^2 + 8 P_1S_1^2 - 2 S_1^4 =0,$$
which can be considered as a quadratic equation in variable $P_1$ giving two solutions. Due to classical condition
$$(a+b)^2 geq 2ab iff S_1^2 geq 2P_1,$$
only one of these solutions is eligible :
$$P_1=(-4+3sqrt{2})S_1^2 iff ab=(-4+3sqrt{2})(a+b)^2 iff (b-sqrt{2}a)(b-frac{sqrt{2}}{2}a)=0$$
whence the result corresponding to the figure.
The parametric equations of the two lines are
$$(a,b)=(p,p sqrt{2}) text{and} (a,b)=(p,p frac{sqrt{2}}{2}), text{for any} p neq 0$$
Due to the symmetry of equations, we have as well, for any $q neq 0$ :
$$(c,d)=(q,q sqrt{2}) text{and} (c,d)=(q,q frac{sqrt{2}}{2}).$$
A quick glance at any of the four equations show that necessarily $q=frac{1}{p}$. We find back in this way all the solutions given by @Claude Leibovici and @A. Pongrácz .
(*) In fact, the fourth equation is a consequence of the first three. Here is why :
First of all, relationship (F) is equivalent to :
$$(G) S_1^4=frac12P_1^2+4P_1S_1^2.$$
As the fourth equation can be written :
$$(H) (S_1^4+2P_1^2-4P_1S_1^2)(S_2^4+2P_2^2-4P_2S_2^2)=25,$$
using (G) in (H), we get :
$$frac52P_1^2 frac52P_2^2=25,$$
which is a tautology due to the fact that $alpha=P_1P_2=-2.$
edited 14 hours ago
answered 23 hours ago
Jean MarieJean Marie
28.9k41949
28.9k41949
add a comment |
add a comment |
This is not an answer but it is too long for a comment.
Looking at this system of equations, I had a very strange feeling (which I cannot explain).
Using a CAS, I solved equations $(1)$, $(2)$, $(3)$ for $a,b,c$ as functions of $d$ and obtained $8$ solutions which are listed below
$$left{a= frac{2}{d},b= frac{sqrt{2}}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{sqrt{2}}{d},b= frac{2}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{2}{d},b=
-frac{sqrt{2}}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{sqrt{2}}{d},b= frac{2}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{1}{d},b= -frac{sqrt{2}}{d},c= -sqrt{2} dright},left{a=
-frac{sqrt{2}}{d},b= -frac{1}{d},c= -sqrt{2} dright},left{a=
-frac{1}{d},b= frac{sqrt{2}}{d},c= sqrt{2} dright},left{a=
frac{sqrt{2}}{d},b= -frac{1}{d},c= sqrt{2} dright}$$
The problem is that, replacing in $(4)$ any of these solutions the resulting equation is $25=25$ !
4
It is not a problem, Claude: you are perfectly right : if $(a,b,c,d)$ is a solution, $(a/k, b/k , ck, dk)$ is clearly a solution as well... Besides, bonne année 2019 !
– Jean Marie
yesterday
@JeanMarie. Good to see you here ! All my best wishes. Thanks for the comment.
– Claude Leibovici
yesterday
This issue is puzzling : the fact that we have an infinite set of solutions means a kind of (algebraic) dependency of this set of equations. But, how to find it, and, in the case such a dependency is found, how can we exploit it ?
– Jean Marie
yesterday
add a comment |
This is not an answer but it is too long for a comment.
Looking at this system of equations, I had a very strange feeling (which I cannot explain).
Using a CAS, I solved equations $(1)$, $(2)$, $(3)$ for $a,b,c$ as functions of $d$ and obtained $8$ solutions which are listed below
$$left{a= frac{2}{d},b= frac{sqrt{2}}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{sqrt{2}}{d},b= frac{2}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{2}{d},b=
-frac{sqrt{2}}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{sqrt{2}}{d},b= frac{2}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{1}{d},b= -frac{sqrt{2}}{d},c= -sqrt{2} dright},left{a=
-frac{sqrt{2}}{d},b= -frac{1}{d},c= -sqrt{2} dright},left{a=
-frac{1}{d},b= frac{sqrt{2}}{d},c= sqrt{2} dright},left{a=
frac{sqrt{2}}{d},b= -frac{1}{d},c= sqrt{2} dright}$$
The problem is that, replacing in $(4)$ any of these solutions the resulting equation is $25=25$ !
4
It is not a problem, Claude: you are perfectly right : if $(a,b,c,d)$ is a solution, $(a/k, b/k , ck, dk)$ is clearly a solution as well... Besides, bonne année 2019 !
– Jean Marie
yesterday
@JeanMarie. Good to see you here ! All my best wishes. Thanks for the comment.
– Claude Leibovici
yesterday
This issue is puzzling : the fact that we have an infinite set of solutions means a kind of (algebraic) dependency of this set of equations. But, how to find it, and, in the case such a dependency is found, how can we exploit it ?
– Jean Marie
yesterday
add a comment |
This is not an answer but it is too long for a comment.
Looking at this system of equations, I had a very strange feeling (which I cannot explain).
Using a CAS, I solved equations $(1)$, $(2)$, $(3)$ for $a,b,c$ as functions of $d$ and obtained $8$ solutions which are listed below
$$left{a= frac{2}{d},b= frac{sqrt{2}}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{sqrt{2}}{d},b= frac{2}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{2}{d},b=
-frac{sqrt{2}}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{sqrt{2}}{d},b= frac{2}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{1}{d},b= -frac{sqrt{2}}{d},c= -sqrt{2} dright},left{a=
-frac{sqrt{2}}{d},b= -frac{1}{d},c= -sqrt{2} dright},left{a=
-frac{1}{d},b= frac{sqrt{2}}{d},c= sqrt{2} dright},left{a=
frac{sqrt{2}}{d},b= -frac{1}{d},c= sqrt{2} dright}$$
The problem is that, replacing in $(4)$ any of these solutions the resulting equation is $25=25$ !
This is not an answer but it is too long for a comment.
Looking at this system of equations, I had a very strange feeling (which I cannot explain).
Using a CAS, I solved equations $(1)$, $(2)$, $(3)$ for $a,b,c$ as functions of $d$ and obtained $8$ solutions which are listed below
$$left{a= frac{2}{d},b= frac{sqrt{2}}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{sqrt{2}}{d},b= frac{2}{d},c=
-frac{d}{sqrt{2}}right},left{a= frac{2}{d},b=
-frac{sqrt{2}}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{sqrt{2}}{d},b= frac{2}{d},c= frac{d}{sqrt{2}}right},left{a=
-frac{1}{d},b= -frac{sqrt{2}}{d},c= -sqrt{2} dright},left{a=
-frac{sqrt{2}}{d},b= -frac{1}{d},c= -sqrt{2} dright},left{a=
-frac{1}{d},b= frac{sqrt{2}}{d},c= sqrt{2} dright},left{a=
frac{sqrt{2}}{d},b= -frac{1}{d},c= sqrt{2} dright}$$
The problem is that, replacing in $(4)$ any of these solutions the resulting equation is $25=25$ !
answered yesterday
Claude LeiboviciClaude Leibovici
119k1157132
119k1157132
4
It is not a problem, Claude: you are perfectly right : if $(a,b,c,d)$ is a solution, $(a/k, b/k , ck, dk)$ is clearly a solution as well... Besides, bonne année 2019 !
– Jean Marie
yesterday
@JeanMarie. Good to see you here ! All my best wishes. Thanks for the comment.
– Claude Leibovici
yesterday
This issue is puzzling : the fact that we have an infinite set of solutions means a kind of (algebraic) dependency of this set of equations. But, how to find it, and, in the case such a dependency is found, how can we exploit it ?
– Jean Marie
yesterday
add a comment |
4
It is not a problem, Claude: you are perfectly right : if $(a,b,c,d)$ is a solution, $(a/k, b/k , ck, dk)$ is clearly a solution as well... Besides, bonne année 2019 !
– Jean Marie
yesterday
@JeanMarie. Good to see you here ! All my best wishes. Thanks for the comment.
– Claude Leibovici
yesterday
This issue is puzzling : the fact that we have an infinite set of solutions means a kind of (algebraic) dependency of this set of equations. But, how to find it, and, in the case such a dependency is found, how can we exploit it ?
– Jean Marie
yesterday
4
4
It is not a problem, Claude: you are perfectly right : if $(a,b,c,d)$ is a solution, $(a/k, b/k , ck, dk)$ is clearly a solution as well... Besides, bonne année 2019 !
– Jean Marie
yesterday
It is not a problem, Claude: you are perfectly right : if $(a,b,c,d)$ is a solution, $(a/k, b/k , ck, dk)$ is clearly a solution as well... Besides, bonne année 2019 !
– Jean Marie
yesterday
@JeanMarie. Good to see you here ! All my best wishes. Thanks for the comment.
– Claude Leibovici
yesterday
@JeanMarie. Good to see you here ! All my best wishes. Thanks for the comment.
– Claude Leibovici
yesterday
This issue is puzzling : the fact that we have an infinite set of solutions means a kind of (algebraic) dependency of this set of equations. But, how to find it, and, in the case such a dependency is found, how can we exploit it ?
– Jean Marie
yesterday
This issue is puzzling : the fact that we have an infinite set of solutions means a kind of (algebraic) dependency of this set of equations. But, how to find it, and, in the case such a dependency is found, how can we exploit it ?
– Jean Marie
yesterday
add a comment |
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2
I would use that $$a^3+b^3=(a+b)(a^2+b^2-ab)$$ etc
– Dr. Sonnhard Graubner
yesterday
1
Your system is a very interesting (and puzzling!). Could you say where you found it ?
– Jean Marie
14 hours ago