# Completing the Square

## How to Complete the square

Given a term of the form

αx2+βx+γ

it is often useful to express it as

α(x+δ)2

This is often useful when solving certain quadratic equations.  In our case it lets us more easily use Laplace Transform Tables (in the table the form (s+a)202 comes up frequently).

Without loss of generality, we will only consider the case where α=1.  If we divide the original equation by through by α and let b=β/α and c=γ/α, we get

x2+bx+c

and our task is to express it as

(x+d)2+e

We start by setting the two terms to be equal to each other

x2 + bx + c = (x+d)2 + e

Expand the right hand side

x2 + bx + c = x2 + 2dx + d2 + e

Equating the coefficients of like powers of x we get

b = 2d,
c = d2 + e
or
d = b/2, and
e = c - d2

##### Example 1: Completing the square

Complete the square for the expression: s2+2s+10

Solution:
The original function is of the form "s2 + bs + c", so b=2, c=10, and

d = b/2 = 1
e = c - d2 = 10 - 1 = 9.

The desired expression is "(s+d)2 + e" or (s+1)2+9

##### Example 2: Completing the square

Complete the square for the expression: x2+4x+29

Solution:
The original function is of the form "x2 + bx + c", so b=4, c=29, and

d = b/2 = 2
e = c - d2 = 29 - 4 = 25.

The desired expression is "(x+d)2 + e" or (x+2)2+25

References