# What is Completing the Square?

Apr 5, 2018

See explanation below

#### Explanation:

When you have a polynomial such as

${x}^{2} + 4 x + 20$

it is sometimes desirable to express it in the form of

${a}^{2} + {b}^{2}$

To do this, we can artificially introduce a constant which allows us to factor a perfect square out of the expression like so:

${x}^{2} + 4 x + 20$

$= {x}^{2} + 4 x + \textcolor{red}{4} - \textcolor{g r e e n}{4} + 20$

Notice that by simultaneously adding and subtracting $4$, we have not changed the value of the expression.

Now we can do this:

$= \left({x}^{2} + 4 x + \textcolor{red}{4}\right) + \left(20 - \textcolor{g r e e n}{4}\right)$

$= {\left(x + 2\right)}^{2} + 16$

$= {\left(x + 2\right)}^{2} + {4}^{2}$

We have "completed the square"!