Question

How do you factor completely `25a^4-49b^2`?

Answer 1

`25a^4-49b^2 = (5a^2-7b)(5a^2+7b)`

Explanation:

Note that both `25a^4 = (5a^2)^2` and `49b^2=(7b)^2` are perfect squares.

The difference of squares identity can be written:

`A^2-B^2=(A-B)(A+B)`

Using this with `A=5a^2` and `B=7b` we find:

`25a^4-49b^2 = (5a^2)^2-(7b)^2`

`color(white)(25a^4-49b^2) = (5a^2-7b)(5a^2+7b)`

There are no simpler factors.