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If $a^4+64=0$ and $a^2 \ne-4a-8$, what is the value of $a^2-4a$?

I tried to add $-16a^2$ to both sides and doing some algebra but it didnt help much.

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    $\begingroup$ If $a^4=-64$, then $a^2=\pm8i$. What is $(\pm8i)!$ supposed to mean? The extension per $\Gamma$ function? $\endgroup$ – Hagen von Eitzen Jul 26 '13 at 21:55
  • $\begingroup$ This is a high school question,I think it is supposed to be simple. $\endgroup$ – guest Jul 26 '13 at 21:58
  • $\begingroup$ By "$!=$", do you mean $\ne$? $\endgroup$ – 6005 Jul 26 '13 at 21:58
  • $\begingroup$ @HagenvonEitzen: I believe $!=$ is supposed to be $\neq$. OP, please confirm. $\endgroup$ – Ross Millikan Jul 26 '13 at 21:58
  • $\begingroup$ Yes it means "not equal".Sorry for that. $\endgroup$ – guest Jul 26 '13 at 21:59
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Note that $a^4+64=(a^2+4a+8)(a^2-4a+8)$. So if one of the items on the right is not $0$, then $\dots$

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  • $\begingroup$ How could you seperate it like that? Yes the answer is -8,thank you. $\endgroup$ – guest Jul 26 '13 at 22:01
  • $\begingroup$ Great!Thanks for the quick solution. $\endgroup$ – guest Jul 26 '13 at 22:04
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    $\begingroup$ You are welcome. You can check the factorization by multiplying out. You can maybe discover it by $a^4+64=(a^2+8)^2 -(4a)^2$, difference of squares. Your adding $16a^2$ to both sides was a great idea, would have worked. $\endgroup$ – André Nicolas Jul 26 '13 at 22:05
  • $\begingroup$ Super @Andre Nicolas . Cannot be quite without congratulating on the answer . $\endgroup$ – Harish Kayarohanam Jul 26 '13 at 22:07
  • $\begingroup$ How can I mark the question as solved? $\endgroup$ – guest Jul 26 '13 at 22:08
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Here's how to find the factorization, it is very similar to completing the square, we have $$\begin{align*}a^4 + 64 &= a^4 + 16a^2 + 64 - 16a^2\\ &= (a^2 + 8)^2 - 16a^2 \\ &= (a^2 - 4a + 8)(a^2 + 4a + 8)\end{align*}$$

See also http://www.artofproblemsolving.com/Wiki/index.php/Sophie_Germain_Identity

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  • $\begingroup$ Thanks for all answers and comments.I spent more than 20 minutes on this question.I think I should sleep :) $\endgroup$ – guest Jul 26 '13 at 22:10

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