Solutions to a polynomial equation with constraint
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I am looking for solutions to the following simple polynomial equation,
$$
x_1^2 + x_2^2 + y_1^2 + y_2^2 = -2 (x_1 x_2 + 3 y_1 y_2)
$$
where $x_i, y_i in mathbb{R}$. Importantly, I would like only solutions for which $y_1, y_2 >0$.
Thus far, I have been unable to find any solutions satisfying this later constraint, despite the fact that there seems to be no simple proof against their existence.
Is there any easy way to find such solutions?
polynomials roots recreational-mathematics
asked Dec 8 '18 at 19:29
B. EneruB. Eneru
376
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add a comment |
$begingroup$
I am looking for solutions to the following simple polynomial equation,
$$
x_1^2 + x_2^2 + y_1^2 + y_2^2 = -2 (x_1 x_2 + 3 y_1 y_2)
$$
where $x_i, y_i in mathbb{R}$. Importantly, I would like only solutions for which $y_1, y_2 >0$.
Thus far, I have been unable to find any solutions satisfying this later constraint, despite the fact that there seems to be no simple proof against their existence.
Is there any easy way to find such solutions?
polynomials roots recreational-mathematics
asked Dec 8 '18 at 19:29
B. EneruB. Eneru
376
$endgroup$
add a comment |
$begingroup$
I am looking for solutions to the following simple polynomial equation,
$$
x_1^2 + x_2^2 + y_1^2 + y_2^2 = -2 (x_1 x_2 + 3 y_1 y_2)
$$
where $x_i, y_i in mathbb{R}$. Importantly, I would like only solutions for which $y_1, y_2 >0$.
Thus far, I have been unable to find any solutions satisfying this later constraint, despite the fact that there seems to be no simple proof against their existence.
Is there any easy way to find such solutions?
polynomials roots recreational-mathematics
asked Dec 8 '18 at 19:29
B. EneruB. Eneru
376
$endgroup$
I am looking for solutions to the following simple polynomial equation,
$$
x_1^2 + x_2^2 + y_1^2 + y_2^2 = -2 (x_1 x_2 + 3 y_1 y_2)
$$
where $x_i, y_i in mathbb{R}$. Importantly, I would like only solutions for which $y_1, y_2 >0$.
Thus far, I have been unable to find any solutions satisfying this later constraint, despite the fact that there seems to be no simple proof against their existence.
Is there any easy way to find such solutions?
polynomials roots recreational-mathematics
polynomials roots recreational-mathematics
asked Dec 8 '18 at 19:29
B. EneruB. Eneru
376
asked Dec 8 '18 at 19:29
B. EneruB. Eneru
376
asked Dec 8 '18 at 19:29
B. EneruB. Eneru
376
asked Dec 8 '18 at 19:29
B. EneruB. Eneru
376
asked Dec 8 '18 at 19:29
B. EneruB. Eneru
376
376
add a comment |
add a comment |
1 Answer
1
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oldest
votes
$begingroup$
Hint.
$$
(x_1+x_2)^2+(y_1+y_2)^2 = -4y_1y_2
$$
now if $y_i > 0$...
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6243516
$endgroup$
$begingroup$
@ShubhamJohri Thanks!
$endgroup$
– Cesareo
Dec 8 '18 at 19:38
$begingroup$
Ahh, it was that easy!! Thanks for your help.
$endgroup$
– B. Eneru
Dec 8 '18 at 19:38
add a comment |
Your Answer
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1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint.
$$
(x_1+x_2)^2+(y_1+y_2)^2 = -4y_1y_2
$$
now if $y_i > 0$...
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6243516
$endgroup$
$begingroup$
@ShubhamJohri Thanks!
$endgroup$
– Cesareo
Dec 8 '18 at 19:38
$begingroup$
Ahh, it was that easy!! Thanks for your help.
$endgroup$
– B. Eneru
Dec 8 '18 at 19:38
add a comment |
$begingroup$
Hint.
$$
(x_1+x_2)^2+(y_1+y_2)^2 = -4y_1y_2
$$
now if $y_i > 0$...
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6243516
$endgroup$
$begingroup$
@ShubhamJohri Thanks!
$endgroup$
– Cesareo
Dec 8 '18 at 19:38
$begingroup$
Ahh, it was that easy!! Thanks for your help.
$endgroup$
– B. Eneru
Dec 8 '18 at 19:38
add a comment |
$begingroup$
Hint.
$$
(x_1+x_2)^2+(y_1+y_2)^2 = -4y_1y_2
$$
now if $y_i > 0$...
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6243516
$endgroup$
Hint.
$$
(x_1+x_2)^2+(y_1+y_2)^2 = -4y_1y_2
$$
now if $y_i > 0$...
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6243516
edited Dec 8 '18 at 19:37
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6243516
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6243516
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6243516
8,6243516
$begingroup$
@ShubhamJohri Thanks!
$endgroup$
– Cesareo
Dec 8 '18 at 19:38
$begingroup$
Ahh, it was that easy!! Thanks for your help.
$endgroup$
– B. Eneru
Dec 8 '18 at 19:38
add a comment |
$begingroup$
@ShubhamJohri Thanks!
$endgroup$
– Cesareo
Dec 8 '18 at 19:38
$begingroup$
Ahh, it was that easy!! Thanks for your help.
$endgroup$
– B. Eneru
Dec 8 '18 at 19:38
$begingroup$
@ShubhamJohri Thanks!
$endgroup$
– Cesareo
Dec 8 '18 at 19:38
$begingroup$
@ShubhamJohri Thanks!
$endgroup$
– Cesareo
Dec 8 '18 at 19:38
$begingroup$
Ahh, it was that easy!! Thanks for your help.
$endgroup$
– B. Eneru
Dec 8 '18 at 19:38
$begingroup$
Ahh, it was that easy!! Thanks for your help.
$endgroup$
– B. Eneru
Dec 8 '18 at 19:38
add a comment |
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