Solutions to a polynomial equation with constraint
$begingroup$
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
$endgroup$
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
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
8,6733516
$endgroup$
$begingroup$
@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
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3031521%2fsolutions-to-a-polynomial-equation-with-constraint%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
8,6733516
$endgroup$
$begingroup$
@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
8,6733516
$endgroup$
$begingroup$
@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
8,6733516
$endgroup$
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
8,6733516
edited Dec 8 '18 at 19:37
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6733516
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6733516
answered Dec 8 '18 at 19:35
CesareoCesareo
8,6733516
8,6733516
$begingroup$
@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 |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3031521%2fsolutions-to-a-polynomial-equation-with-constraint%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
