COMPLETING THE SQUARE
KEVIN D. KING
1. How to factor x2 + bx + c = 0
We can solve quadratic equations by a method known as Completing the Square.
Let’s start with an example equation,
x2 + 6x + 10 = 0
To begin, subtract the constant term from each side to obtain x2 + 6x = −10
The Addition Property of Equality states that you can add the same term to
both sides, so:
x2 + 6x + (something) = −10 + (something)
Next, you consider what that ”something” should be. To do this, take your b
coefficient, divide it by two and square it. That is the ”something” that should be
added to each side. For example, 6 ÷ 2 = 3 and 32 = 9.
From there, you can factor one side of your equation as a perfect square. You
then proceed to solve by taking the square root of each side.
x2 + 6x + 9 = −10 + 9
(x + 3)2 = −1
√
x + 3 = ± −1
x + 3 = ±i
x = −3 ± i
Note that we make use of the complex number i which is the square root of
−1. Thus the original equation has no real roots. The roots are instead a pair of
complex conjugates.
Generally, you repeat the steps above to solve x2 + bx + c = 0 for the variable x.
If you instead want to solve ax2 + bx + c = 0 for the variable x, divide both sides of
your equation by coefficient a first to obtain a new equation (with new coefficients)
where a = 1.
Date: January 5, 2008.
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KEVIN D. KING
2. The Quadratic Formula
Suppose you want to solve ax2 + bx + c = 0 where a 6= 0. We solve this equation for x by completing the square. What results is a formula that allows you to
immediately solve for x without factoring altogether.
Please keep in mind that you are still required to know BOTH the quadratic formula and how to complete the square for the quiz and test. Knowing how to derive
the quadratic formula in this manner is a possible bonus question.
ax2 + bx + c = 0
b
c
0
x2 + x + =
a
a
a
b
0
c
x2 + x = −
a
a a
b2
c
b2
b
x2 + x + 2 = − + 2
a
4a
a 4a
b
b2
b2
4ac
2
x + x+ 2 = 2 − 2
a
4a
4a
4a
b
b2
b2 − 4ac
2
x + x+ 2 =
a
4a
4a2
2
b
b − 4ac
(x + )2 =
2
2a
r 4a
b2 − 4ac
b
x+
=±
2
2a
√ 4a
2
b − 4ac
b
=±
x+
2a
√ 2a
−b ± b2 − 4ac
x=
2a
This last line is known as the Quadratic Formula, which helps find roots for all
equations of the form ax2 + bx + c = 0 where a, b, c ∈ R and a 6= 0.