# When do I know when to use "completing the square"?

Jun 15, 2015

It depends on what information you are trying to get and how simple the quadratic problem you are facing is...

#### Explanation:

If you are trying to find the vertex of a parabola described by a quadratic equation, then completing the square is the most natural way to do it.

If you are trying to find the roots of a quadratic equation, then completing the square will 'always work', in the sense that it does not require the factors to be rational and in the sense that it will give you the complex roots if the quadratic's roots are not real.

On the other hand, there may be obvious or easy to find factorings that are a little quicker.

For example, suppose you are trying to factorize the quadratic:

$f \left(x\right) = 37 {x}^{2} - 13 x - 24$

It looks a little tedious to do, but notice that the sum of the coefficients ($37 - 13 - 24$) is $0$. That means that $f \left(1\right) = 0$ and $\left(x - 1\right)$ is a factor of $f \left(x\right)$. It's then easy to find the other factor:

$37 {x}^{2} - 13 x - 24 = \left(x - 1\right) \left(37 x + 24\right)$

If a quadratic is obviously of the form ${a}^{2} + 2 a b + {b}^{2}$ then I know it's already square, being equal to ${\left(a + b\right)}^{2}$. For example:

$9 {x}^{2} - 24 x + 16 = {\left(3 x - 4\right)}^{2}$ with $a = 3 x$ and $b = - 4$.

In general you can complete the square as follows:

$a {x}^{2} + b x + c = a {\left(x + \frac{b}{2 a}\right)}^{2} + \left(c - {b}^{2} / \left(4 a\right)\right)$

I usually first check $\Delta = {b}^{2} - 4 a c$ to see if I'm facing a quadratic that will factor nicely or I have to use heavier methods.