Why is $(5\sqrt{5p}-3\sqrt{5q})(5\sqrt{5p}+3\sqrt{5q}) \equiv 5(5p-3q)(5p+3q)$?

Question

I was working on the difference of two squares, $125p^2-45q^2$

Writing my answer, $$(5\sqrt{5}p-3\sqrt{5}q)(5\sqrt{5}p+3\sqrt{5}q),$$ onto Pearson, I got a popup that said my answer was equivalent to the correct answer but in incorrect form. Apparently, the correct answer is $$5(5p-3q)(5p+3q).$$

Why is that? I'm failing to see the intuition here.

Answer 1

$$125p^2-45q^2=5(25p^2-9q^2)=5(5p+3q)(5p-3q)$$ Your answer was incorrect because it should have been $$125p^2-45q^2=(\sqrt{125}p^2+\sqrt{45}q^2)(\sqrt{125}p^2-\sqrt{45}q^2)=(5\sqrt{5}p+3\sqrt{5}q)(5\sqrt{5}p-3\sqrt{5}q)=5(5p+3q)(5p-3q)$$

Answer 2

Think like this:

$$ 125p^2-45q^2= 5\cdot 5^2p^2- 5\cdot 3^2q^2= 5\big((5p)^2-(3q)^2\big)=5(5p-3q)(5p+3q) $$

This way you skip dealing with those pesky squareroots.

Hope this helps :)

Answer 3

Your answer is not correct or Person's algebraic module is buggy.

Your supposed equivalent expression should be $$(5p\sqrt{5}-3q\sqrt{\color{red}{5}})(5p\sqrt{5}+3q\sqrt{\color{red}{5}})$$

Edit after OP's edit:

Note that

• $125p^2 = \left(5\sqrt{5}\cdot p\right)^2$
• $45q^2 = \left(3\sqrt{5}\cdot q\right)^2$