# Have trouble reducing ((a + c)^2 - b^2) / (4a^2c^2 - (a^2 + c^2 - b^2)^2)

I'm trying to reduce the below equation but I'm kind of stuck. This is what I have done so far.

$((a + c)^2 - b^2) / (4a^2c^2 - (a^2 + c^2 - b^2)^2)$

--> $(a + c - b) (a + c + b) / (4a^2c^2 - a^4 - 2a^2c^2 + 2a^2b^2 - c^4 + 2b^2c^2 - b^4)$

--> $(a + c - b) (a + c + b) / (2a^2c^2 - a^4 - + 2a^2b^2 - c^4 + 2b^2c^2 - b^4)$

--> $(a + c - b) (a + c + b) / (a^2(2c^2 - a^2 + 2b^2) + b^2(2c^2 - b^2) - c^4)$

??????

This equation seems to grow bigger and bigger and I'm not able to reduce it. Did I do something wrong?

Thank you in advance

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Yes, you did something wrong. The given expression has lots of structure. By multiplying out you are destroying visual structure. The bottom is a difference of two squares. Exploit that! –  André Nicolas Nov 19 '11 at 1:29

## 1 Answer

Notice the denominator is the difference of two squares just like the numerator. Indeed,

$$\color{Red}{4a^2c^2}-\color{Blue}{(a^2+c^2-b^2)^2}=\left(\color{Red}{2ac}-\color{Blue}{(a^2+c^2-b^2)}\right)\left(\color{Red}{2ac}+\color{Blue}{(a^2+c^2-b^2)}\right).$$

Now use $(a\pm c)^2=\color{Blue}{a^2}\pm \color{Red}{2ac}+\color{Blue}{c^2}$ and then do a simple cancellation.

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Thank you. I solved it. –  yyc2001 Nov 19 '11 at 2:00