Solve Quadratic Equations by Completing the Square
Goals • Solve quadratic equations by completing
the square.
Your Notes
VOCABULARY
Completing the square Adding a constant c to an
expression of the form x^{2} + bx to form a perfect
square trinomial
COMPLETING THE SQUARE
Words
To complete the square for the expression
x^{2} + bx, add the square of half the coefficient
of x.
Algebra
Example 1 Complete the square
Find the value of c that makes the expression
x^{2}  5x + c a perfect square trinomial. Then write
the expression as the square of the binomial.
Solution
Step 1 Find the value of c.
Find the square of half the
coefficient of x.
Step 2 Write the expression as a perfect
square trinomial.
Then write the expression as the square of a
binomial.
Substitute
for c.
Square of a binomial
● Guided Practice Find the value of c that makes the
expression a perfect square trinomial. Then write
the expression as the square of a binomial.
Example 2 Multiple Choice Practice
What quantity should be added to both sides of this
equation to complete the square?
x^{2}  16x = 10
A 64
B 8
C 8
D 64
Solution
Find the square of half of the coefficient of x:
The correct answer is D . A B C D
Example 3 Solve a quadratic equation
Solve 4x^{2}  16x + 8 = 0 by completing the square.
Solution


Write original
equation. 


Subtract 8
from each side. 
Be sure that the
coefficient of x^{2}
is 1 before you
complete the
square. 

Divide each side
by 4 . 


Add 


or , to
each side. 


Write left side as
the square of a
binomial. 


Take square roots
of each side. 


Add 2 to
each side. 

The solutions are 
Check To check the solutions
of 4x^{2} 16x + 8 = 0, graph
the related function.
The xintercepts are
approximately 0.6 and 3.4.
Compare these values with
the solutions:
Homework
