Question 203870
{{{x^4 - 10x^2 = -24}}}
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let z = {{{x^2}}}
equation becomes:
{{{z^2 - 10z = -24}}}
add 24 to both sides of this equation to get:
{{{x^2 - 10z + 24 = 0}}}
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This can be factored as follows:
{{{(z-6) * (z-4) = 0}}}
your possible solutions are:
z = 6
z = 4
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if you  let z = 6, then {{{z^2 - 10z + 24}}} = 36 - 60 + 24 = 60 - 60 = 0 so z = 6 is a good answer.
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if you let z = 4, then {{{z^2 - 10z + 24}}} = 16 - 40 + 24 = 40 - 40 = 0 so z = 4 is a good answer.
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this shows that your z equation has good answers.
you now need to go back to your x equation and see what answers are good for that.
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since you let z = x^2, this means that your possible solutions for x are:
{{{x^2}}} = 6
{{{x^2}}} = 4
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if you take the square root of both sides of these equations you get:
x = +/- {{{sqrt(6)}}}
x = +/- {{{sqrt(4)}}}
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you have 4 possible answers to your problem.
x = {{{sqrt(6)}}}
x = - {{{sqrt(6)}}}
x = {{{sqrt(4)}}} = 2
x = - {{{sqrt(4)}}} = -2
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you need to plug them into the original equation to see which ones fit.
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your original equation is {{{x^4 - 10x^2 + 24 = 0}}} after you added 24 to both sides of it which we did in the beginning.
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let x = {{{sqrt(6)}}}
{{{x^4 - 10x^2 + 24}}} = 36 - 60 + 24 = 0
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let x = (- {{{sqrt(6)}}})
{{{x^4 - 10x^2 + 24}}} = {{{(-sqrt(6))^4 - 10*(-sqrt(6))^2 + 24}}}
which equals 36 - 60 + 24 = 0
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let x = {{{sqrt(4)}}} = 2
{{{x^4 - 10x^2 + 24}}} = {{{2^4 - 10*2^2 + 24}}} = 16 - 40 + 24 = 0
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let x = (- {{{sqrt(4)}}} = -2
{{{x^4 - 10x^2 + 24}}} = {{{(-2)^4 - 10*(-2)^2 + 24}}} = 16 - 40 + 24 = 0
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looks like all values of x are good.
your answer would be:
x = +/- {{{sqrt(6)}}}
and
x = +/- 2
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