Question

Which of the following is the correct factored form of the given equation?

Answer 1

Hint: "4x 2 - 25 = 0" is nearly meaningless. You want 4x^2 - 25 = 0.

Note that both 4x^2 and 25 are perfect squares, so this is the difference of two squares, and is equal to (2x-5)(2x+5)= 0.

Answer 2

There is a small subset of polynomials whos factorizations mathematicians like to label "special." From an outside perspective, nothing seems to be different about these polynomials. You can use the same algorithm you use to find any other factorization, but you could also form a general statement about polynomials like this.

This polynomial is in the form a^2-b^2. We can factor this normally.

-b and b add to 0 and multiply to -b^2, so:

$a^2+ab-ab-b^2=\\ a(a+b)-b(a-b)=\\ (a-b)(a+b)$

So, we've factored this quadratic like we would any other, but we also notice something interesting about all quadratics in this form. In this example,

$a^2=4x^2\\ b^2=25$

That means

$a=2x\\ b=5$

So we can say that the factorization of the quadratic is:

$(2x-5)(2x+5)$