How do we solve for all values of $x$?
$begingroup$
Suppose
$$(x^2 - 5)^{8} (x+1)^{-16} = 1$$
How do we solve for all values of $x$?
I think we can equate the bases to $1$ in order to get $$1 times 1 = 1$$
$$x^2-5 = 1, x^2-5=-1$$
or
$$x + 1 = 1, x + 1 = -1$$
Could you assist me with this?
algebra-precalculus
$endgroup$
add a comment |
$begingroup$
Suppose
$$(x^2 - 5)^{8} (x+1)^{-16} = 1$$
How do we solve for all values of $x$?
I think we can equate the bases to $1$ in order to get $$1 times 1 = 1$$
$$x^2-5 = 1, x^2-5=-1$$
or
$$x + 1 = 1, x + 1 = -1$$
Could you assist me with this?
algebra-precalculus
$endgroup$
add a comment |
$begingroup$
Suppose
$$(x^2 - 5)^{8} (x+1)^{-16} = 1$$
How do we solve for all values of $x$?
I think we can equate the bases to $1$ in order to get $$1 times 1 = 1$$
$$x^2-5 = 1, x^2-5=-1$$
or
$$x + 1 = 1, x + 1 = -1$$
Could you assist me with this?
algebra-precalculus
$endgroup$
Suppose
$$(x^2 - 5)^{8} (x+1)^{-16} = 1$$
How do we solve for all values of $x$?
I think we can equate the bases to $1$ in order to get $$1 times 1 = 1$$
$$x^2-5 = 1, x^2-5=-1$$
or
$$x + 1 = 1, x + 1 = -1$$
Could you assist me with this?
algebra-precalculus
algebra-precalculus
asked Dec 18 '18 at 21:35
EnzoEnzo
19917
19917
add a comment |
add a comment |
3 Answers
3
active
oldest
votes
$begingroup$
$(x^2 - 5)^{8} (x+1)^{-16} = 1$ means
$(x^2 - 5)^{8} = (x+1)^{16}$.
Assuming you only want real and not complex answers then
$ sqrt[8]{(x^2 - 5)^{8}} = sqrt[8]{(x+1)^{16}}$
$|x^2 - 5| = (x+1)^2$
or either A) $x^2 - 5 = (x+1)^2$ or B) $5- x^2 = (x+1)^2$
A) $x^2 - 5 = x^2 + 2x + 1$
$2x = -6$
$x = -3$.
Verification: $(x^2 - 5)^8(x+1)^{-16}=frac{(9-5)^8}{(-2)^{16}}= frac {4^8}{2^{16}}=frac {2^{16}}{2^{16}} = 1$.
B)$5- x^2 = (x+1)^2$
$5-x^2 = x^2 + 2x + 1$
$2x^2 + 2x -4 = 0$
$x^2 + x -2 = 0$
$(x +2)(x-1) = 0$ so either C) $x+2 = 0$ and $x=-2$ or $x-1=0$ and $x = 1$.
So solutions are $x = -3; x = -2;$ or $x = 1$.
Verification:
$(x^2 - 5)^8(x+1)^{-16} = frac {(4-5)^8}{(-2 + 1)^{16}}= frac {(-1)^8}{(-1)^{16}} = frac 11 = 1$. so $x = -2$ is a solution.
$(x^2 - 5)^8(x+1)^{-16} = frac {(1-5)^8}{(1 + 1)^{16}}= frac {(-4)^8}{(2)^{16}} = frac {4^8}{4^8} = 1$. so $x = 1$ is a solution.
$endgroup$
add a comment |
$begingroup$
We have $(x^2-5)^8(x+1)^{-16}=1$, so $left(frac{x^2-5}{(x+1)^2}right)^8=1$, so $frac{x^2-5}{(x+1)^2}$ is an $8$'th root of unity. Assuming you need to solve over the reals, we get $frac{x^2-5}{(x+1)^2}in{-1,1}$. Fixing for $-1$ or $1$, you can multiply both sides by $(x+1)^2$ to get a quadratic equation.
$endgroup$
add a comment |
$begingroup$
$(x^2-5)^8=(x+1)^{16}$
So
$[(x^2-5)^4-(x+1)^8] [(x^2-5)^4+(x+1)^8]=0$
If and only if
$[(x^2-5)^4-(x+1)^8]=0$
If and only if
$[(x^2-5)^2-(x+1)^4]=0$
If and only if
$[(x^2-5)-(x+1)^2] [(x^2-5)+(x+1)^2]=0$
So
$[-2x-6][2x^2+2x-4]=0$
$endgroup$
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%2f3045737%2fhow-do-we-solve-for-all-values-of-x%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
3 Answers
3
active
oldest
votes
3 Answers
3
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
$(x^2 - 5)^{8} (x+1)^{-16} = 1$ means
$(x^2 - 5)^{8} = (x+1)^{16}$.
Assuming you only want real and not complex answers then
$ sqrt[8]{(x^2 - 5)^{8}} = sqrt[8]{(x+1)^{16}}$
$|x^2 - 5| = (x+1)^2$
or either A) $x^2 - 5 = (x+1)^2$ or B) $5- x^2 = (x+1)^2$
A) $x^2 - 5 = x^2 + 2x + 1$
$2x = -6$
$x = -3$.
Verification: $(x^2 - 5)^8(x+1)^{-16}=frac{(9-5)^8}{(-2)^{16}}= frac {4^8}{2^{16}}=frac {2^{16}}{2^{16}} = 1$.
B)$5- x^2 = (x+1)^2$
$5-x^2 = x^2 + 2x + 1$
$2x^2 + 2x -4 = 0$
$x^2 + x -2 = 0$
$(x +2)(x-1) = 0$ so either C) $x+2 = 0$ and $x=-2$ or $x-1=0$ and $x = 1$.
So solutions are $x = -3; x = -2;$ or $x = 1$.
Verification:
$(x^2 - 5)^8(x+1)^{-16} = frac {(4-5)^8}{(-2 + 1)^{16}}= frac {(-1)^8}{(-1)^{16}} = frac 11 = 1$. so $x = -2$ is a solution.
$(x^2 - 5)^8(x+1)^{-16} = frac {(1-5)^8}{(1 + 1)^{16}}= frac {(-4)^8}{(2)^{16}} = frac {4^8}{4^8} = 1$. so $x = 1$ is a solution.
$endgroup$
add a comment |
$begingroup$
$(x^2 - 5)^{8} (x+1)^{-16} = 1$ means
$(x^2 - 5)^{8} = (x+1)^{16}$.
Assuming you only want real and not complex answers then
$ sqrt[8]{(x^2 - 5)^{8}} = sqrt[8]{(x+1)^{16}}$
$|x^2 - 5| = (x+1)^2$
or either A) $x^2 - 5 = (x+1)^2$ or B) $5- x^2 = (x+1)^2$
A) $x^2 - 5 = x^2 + 2x + 1$
$2x = -6$
$x = -3$.
Verification: $(x^2 - 5)^8(x+1)^{-16}=frac{(9-5)^8}{(-2)^{16}}= frac {4^8}{2^{16}}=frac {2^{16}}{2^{16}} = 1$.
B)$5- x^2 = (x+1)^2$
$5-x^2 = x^2 + 2x + 1$
$2x^2 + 2x -4 = 0$
$x^2 + x -2 = 0$
$(x +2)(x-1) = 0$ so either C) $x+2 = 0$ and $x=-2$ or $x-1=0$ and $x = 1$.
So solutions are $x = -3; x = -2;$ or $x = 1$.
Verification:
$(x^2 - 5)^8(x+1)^{-16} = frac {(4-5)^8}{(-2 + 1)^{16}}= frac {(-1)^8}{(-1)^{16}} = frac 11 = 1$. so $x = -2$ is a solution.
$(x^2 - 5)^8(x+1)^{-16} = frac {(1-5)^8}{(1 + 1)^{16}}= frac {(-4)^8}{(2)^{16}} = frac {4^8}{4^8} = 1$. so $x = 1$ is a solution.
$endgroup$
add a comment |
$begingroup$
$(x^2 - 5)^{8} (x+1)^{-16} = 1$ means
$(x^2 - 5)^{8} = (x+1)^{16}$.
Assuming you only want real and not complex answers then
$ sqrt[8]{(x^2 - 5)^{8}} = sqrt[8]{(x+1)^{16}}$
$|x^2 - 5| = (x+1)^2$
or either A) $x^2 - 5 = (x+1)^2$ or B) $5- x^2 = (x+1)^2$
A) $x^2 - 5 = x^2 + 2x + 1$
$2x = -6$
$x = -3$.
Verification: $(x^2 - 5)^8(x+1)^{-16}=frac{(9-5)^8}{(-2)^{16}}= frac {4^8}{2^{16}}=frac {2^{16}}{2^{16}} = 1$.
B)$5- x^2 = (x+1)^2$
$5-x^2 = x^2 + 2x + 1$
$2x^2 + 2x -4 = 0$
$x^2 + x -2 = 0$
$(x +2)(x-1) = 0$ so either C) $x+2 = 0$ and $x=-2$ or $x-1=0$ and $x = 1$.
So solutions are $x = -3; x = -2;$ or $x = 1$.
Verification:
$(x^2 - 5)^8(x+1)^{-16} = frac {(4-5)^8}{(-2 + 1)^{16}}= frac {(-1)^8}{(-1)^{16}} = frac 11 = 1$. so $x = -2$ is a solution.
$(x^2 - 5)^8(x+1)^{-16} = frac {(1-5)^8}{(1 + 1)^{16}}= frac {(-4)^8}{(2)^{16}} = frac {4^8}{4^8} = 1$. so $x = 1$ is a solution.
$endgroup$
$(x^2 - 5)^{8} (x+1)^{-16} = 1$ means
$(x^2 - 5)^{8} = (x+1)^{16}$.
Assuming you only want real and not complex answers then
$ sqrt[8]{(x^2 - 5)^{8}} = sqrt[8]{(x+1)^{16}}$
$|x^2 - 5| = (x+1)^2$
or either A) $x^2 - 5 = (x+1)^2$ or B) $5- x^2 = (x+1)^2$
A) $x^2 - 5 = x^2 + 2x + 1$
$2x = -6$
$x = -3$.
Verification: $(x^2 - 5)^8(x+1)^{-16}=frac{(9-5)^8}{(-2)^{16}}= frac {4^8}{2^{16}}=frac {2^{16}}{2^{16}} = 1$.
B)$5- x^2 = (x+1)^2$
$5-x^2 = x^2 + 2x + 1$
$2x^2 + 2x -4 = 0$
$x^2 + x -2 = 0$
$(x +2)(x-1) = 0$ so either C) $x+2 = 0$ and $x=-2$ or $x-1=0$ and $x = 1$.
So solutions are $x = -3; x = -2;$ or $x = 1$.
Verification:
$(x^2 - 5)^8(x+1)^{-16} = frac {(4-5)^8}{(-2 + 1)^{16}}= frac {(-1)^8}{(-1)^{16}} = frac 11 = 1$. so $x = -2$ is a solution.
$(x^2 - 5)^8(x+1)^{-16} = frac {(1-5)^8}{(1 + 1)^{16}}= frac {(-4)^8}{(2)^{16}} = frac {4^8}{4^8} = 1$. so $x = 1$ is a solution.
answered Dec 18 '18 at 21:56
fleabloodfleablood
73.6k22891
73.6k22891
add a comment |
add a comment |
$begingroup$
We have $(x^2-5)^8(x+1)^{-16}=1$, so $left(frac{x^2-5}{(x+1)^2}right)^8=1$, so $frac{x^2-5}{(x+1)^2}$ is an $8$'th root of unity. Assuming you need to solve over the reals, we get $frac{x^2-5}{(x+1)^2}in{-1,1}$. Fixing for $-1$ or $1$, you can multiply both sides by $(x+1)^2$ to get a quadratic equation.
$endgroup$
add a comment |
$begingroup$
We have $(x^2-5)^8(x+1)^{-16}=1$, so $left(frac{x^2-5}{(x+1)^2}right)^8=1$, so $frac{x^2-5}{(x+1)^2}$ is an $8$'th root of unity. Assuming you need to solve over the reals, we get $frac{x^2-5}{(x+1)^2}in{-1,1}$. Fixing for $-1$ or $1$, you can multiply both sides by $(x+1)^2$ to get a quadratic equation.
$endgroup$
add a comment |
$begingroup$
We have $(x^2-5)^8(x+1)^{-16}=1$, so $left(frac{x^2-5}{(x+1)^2}right)^8=1$, so $frac{x^2-5}{(x+1)^2}$ is an $8$'th root of unity. Assuming you need to solve over the reals, we get $frac{x^2-5}{(x+1)^2}in{-1,1}$. Fixing for $-1$ or $1$, you can multiply both sides by $(x+1)^2$ to get a quadratic equation.
$endgroup$
We have $(x^2-5)^8(x+1)^{-16}=1$, so $left(frac{x^2-5}{(x+1)^2}right)^8=1$, so $frac{x^2-5}{(x+1)^2}$ is an $8$'th root of unity. Assuming you need to solve over the reals, we get $frac{x^2-5}{(x+1)^2}in{-1,1}$. Fixing for $-1$ or $1$, you can multiply both sides by $(x+1)^2$ to get a quadratic equation.
edited Dec 18 '18 at 22:46
answered Dec 18 '18 at 21:47
SmileyCraftSmileyCraft
3,761519
3,761519
add a comment |
add a comment |
$begingroup$
$(x^2-5)^8=(x+1)^{16}$
So
$[(x^2-5)^4-(x+1)^8] [(x^2-5)^4+(x+1)^8]=0$
If and only if
$[(x^2-5)^4-(x+1)^8]=0$
If and only if
$[(x^2-5)^2-(x+1)^4]=0$
If and only if
$[(x^2-5)-(x+1)^2] [(x^2-5)+(x+1)^2]=0$
So
$[-2x-6][2x^2+2x-4]=0$
$endgroup$
add a comment |
$begingroup$
$(x^2-5)^8=(x+1)^{16}$
So
$[(x^2-5)^4-(x+1)^8] [(x^2-5)^4+(x+1)^8]=0$
If and only if
$[(x^2-5)^4-(x+1)^8]=0$
If and only if
$[(x^2-5)^2-(x+1)^4]=0$
If and only if
$[(x^2-5)-(x+1)^2] [(x^2-5)+(x+1)^2]=0$
So
$[-2x-6][2x^2+2x-4]=0$
$endgroup$
add a comment |
$begingroup$
$(x^2-5)^8=(x+1)^{16}$
So
$[(x^2-5)^4-(x+1)^8] [(x^2-5)^4+(x+1)^8]=0$
If and only if
$[(x^2-5)^4-(x+1)^8]=0$
If and only if
$[(x^2-5)^2-(x+1)^4]=0$
If and only if
$[(x^2-5)-(x+1)^2] [(x^2-5)+(x+1)^2]=0$
So
$[-2x-6][2x^2+2x-4]=0$
$endgroup$
$(x^2-5)^8=(x+1)^{16}$
So
$[(x^2-5)^4-(x+1)^8] [(x^2-5)^4+(x+1)^8]=0$
If and only if
$[(x^2-5)^4-(x+1)^8]=0$
If and only if
$[(x^2-5)^2-(x+1)^4]=0$
If and only if
$[(x^2-5)-(x+1)^2] [(x^2-5)+(x+1)^2]=0$
So
$[-2x-6][2x^2+2x-4]=0$
answered Dec 18 '18 at 21:40
Federico FalluccaFederico Fallucca
2,280210
2,280210
add a comment |
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%2f3045737%2fhow-do-we-solve-for-all-values-of-x%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