Is it possible to solve this quadratic equation without calculating discriminant?

Question

I have the quadratic equation $\;\;5x^2+96x-576=0\;\;$. I wonder can we solve it without using formula $x=\frac{-b\pm\sqrt{\Delta}}{2a}$ ? I suspect there is some way to do it because we have a lot of $24$s , ( $96=24\times4$ and $576=24^2)$ but I can't find it.

Answer 1

Let $y = 24$ and you have

$$5x^2+4xy - y^2$$

which easily factors as $(5x-y)(x+y)$.

Answer 2

On recognizing that $96=24\cdot4$ and $576=24^2$, it makes sense to let $x=24u$, factor out the $24^2$ from all three terms, and reduce the quadratic to

$$5u^2+4u-1=0$$

This factors easily into $(5u-1)(u+1)=0$, at which point you can let $u=x/24$, multiply the $24^2$ back in, and get

$$(5x-24)(x+24)=0$$

Answer 3

An easy method is to start with factoring out $x$ and $24$

$$5x^2+96x-576=0$$ $$(5x^2-24)+(120x-576)\tag{Break into groups}$$ $$x(5x-24)+24(5x-24)\tag{Factor out x and 24}$$ $$(5x-24)(x+24)=0$$