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I am trying to simplify this

$$\frac{9x^2 - x^4} {x^2 - 6x +9}$$

The solution is

$$\frac{-x^2(x +3)}{x-3} = \frac{-x^3 - 3x^2}{x-3} $$

I have done $$\frac{x^2(9-x^2)}{(x-3)(x-3)} = \frac{x^2(3-x)(3+x)}{(x-3)(x-3)} $$

but I do not find a way to simplify

How can I simplify to get the answer?

I have to use

$${(a +b)(a -b)} = {a^2 - b^2} $$ $${(a +b)^2} = {a^2 + 2ab+b^2} $$ $${(a -b)^2} = {a^2 - 2ab+b^2} $$

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  • $\begingroup$ You have done all the hard work! $\endgroup$ Jan 27, 2015 at 23:54

2 Answers 2

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Notice that $(3-x)=-(x-3)$. Simplify accordingly.

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  • $\begingroup$ Oh I did not see that , x^2 - (x-3)(x +3)/(x -3)(x-3), so I can obtain the answer, thanks for answering $\endgroup$
    – Learner
    Jan 27, 2015 at 23:50
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Observe that $$ 9x^2-x^4=-x^2(x^2-9)=-x^2(x-3)(x+3). $$ Thus, we have that $$ \frac{9x^2-x^4}{x^2-6x+9}=\frac{-x^2(x-3)(x+3)}{(x-3)^2}=-x^2\left(\frac{x+3}{x-3}\right). $$

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  • $\begingroup$ Thanks for answering I did not see that until the first answer $\endgroup$
    – Learner
    Jan 27, 2015 at 23:53
  • $\begingroup$ Which is wrong, since $x^2-6x+9=(x-3)^2$ and not $(x-3)(x+3)$ $\endgroup$
    – Demosthene
    Jan 27, 2015 at 23:53
  • $\begingroup$ @Learner My initial answer was wrong. I've corrected it now (what Demosthene pointed out was right). $\endgroup$ Jan 27, 2015 at 23:55
  • $\begingroup$ @induktio Better :) $\endgroup$
    – Demosthene
    Jan 27, 2015 at 23:55
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    $\begingroup$ @Demosthene Thanks for pointing it out. *tips fedora* $\endgroup$ Jan 27, 2015 at 23:56

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