How to factor $a^2+10ab+25b^2-9c^2$

Question

This is what I got so far $a^2+10ab+(5b)^2-(3c)^2$. I think I can group $a^2+10ab$ but not sure about $(5b)^2-(3c)^2$. Can someone explain this?

Answer 1

Hint: $a^2+10ab+25b^2=(a+5b)^2$

Answer 2

Write $10ab=2(a)(5b)$. Do you see now how you can factor $a^2+10ab+25b^2$? That, and the difference of squares formula, will get you the rest of the way.

Answer 3

$$a^2+10ab+(5b)^2-(3c)^2 = a^2+10ab+25b^2 - (3c)^2 =(a+5b)^2 - (3c)^2$$

Now we have a difference of squares, which factors very nicely:

$$(a+5b)^2 - (3c)^2 = (a + 5b + 3c)(a+5b - 3c)$$

Answer 4

When factoring an expression, one of the things that you want to notice right away is to find a term that contains a cross term, as in, having multiple variables.

In your case it is the $10ab$.

So, noticing that $25b^2 - 9c^2 = (5b-3c)(5b+3c)$ IS an important thing, and good for you to be able to know that.

However, the $10ab$ will get in your way, so you want to split the $a \text{ and } b $ somehow. That's the reason why it is most natural to try factoring $a^2+10ab+25b^2$ after noticing that your first trial did not work.