Given a term of the form

αx^{2}+βx+γ

it is often useful to express it as

α(x+δ)^{2}+γ

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)^{2}+ω_{0}^{2} 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

x^{2}+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

x^{2} + bx + c = (x+d)^{2} + e

Expand the right hand side

x^{2} + bx + c = x^{2} + 2dx + d^{2} + e

Equating the coefficients of like powers of x we get

b = 2d,

c = d^{2} + e

or

d = b/2, and

e = c - d^{2}

Complete the square for the expression: s^{2}+2s+10

**Solution**:

The original
function is of the form "s^{2} + bs + c", so b=2, c=10, and

d = b/2 = 1

e = c - d^{2} = 10 - 1 = 9.

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

Complete the square for the expression: x^{2}+4x+29

**Solution**:

The original
function is of the form "x^{2} + bx + c", so b=4, c=29, and

d = b/2 = 2

e = c - d^{2} = 29 - 4 = 25.

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

© Copyright 2005 to 2019 Erik Cheever This page may be freely used for educational purposes.

Erik Cheever Department of Engineering Swarthmore College