How do I quickly factorize quadratic equations?

Question

Whenever I have to factorize an equation I usually just look for the common factors and then just work form there. However, I was wondering whether there is a quicker way to get the factorized form.

Edit

How to quickly factorize quadratic equations of the form $$ ax^2 + by + c = 0, $$ where $a,b,c\in\Bbb{R}$.

Simple example: Factorize $x^2 + 3x - 4 = 0$

Help would be appreciated.

Thank you :)

Answer 1

The best way, I call it middle-split.

Taking your equation under the spotlight

$x^2+3x-4=0$

Think of 2 numbers $p,q$ such that $p+q=3$ and $pq=-4$

That yields $4,-1$ as one of the pair.

So, we split the equation from the middle such that $x^2-x+4x-4=0$

Now, it is easy to factorize. $x(x-1)+4(x-1)=0$

$(x+4)(x-1)=0$

So, either $x=-4$ or $x=1$

Quite often this trick will work. But, always emphasis that you use the quadratic formula, which removes the "guess-work" involved, and relies completely on computation.

$$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$

Answer 2

For $x^2+3x-4$, it should be obvious that $4\times(-1)=-4$, and $4+(-1)=3$. Basically you want to look for two numbers $x,y$ such that $x\times y$ gives you the coefficient of $x$, and $x+y$ gives you the coefficient of the constant. The only way to be quick at this is to practise a lot.

Also, use WolframAlpha when you are sleepy.