Download Solving a Quadratic Equation

5.5
Completing the Square
Goals p Solve quadratic equations by completing the square.
p Write quadratic functions in vertex form.
Your Notes
VOCABULARY
Completing the square The process that allows you to
write an expression of the form x 2 bx as the square
of a binomial
Example 1
Completing the Square
Find the value of c that makes x2 12x c a perfect
square trinomial, then write the expression as the square of
a binomial.
Solution
In the expression x2 12x c, notice that b 12 .
Therefore:
b 2
12 2
c 36
2
2
冢 冣
冢 冣
Use this value of c to write x2 12x c as a perfect square
trinomial, and then as the square of a binomial.
x2 12x c x2 12x 36
(x 6)2
Perfect square trinomial
Square of binomial:
冢x ⴙ 2冣
b 2
Checkpoint Complete the following exercise.
1. Find the value of c that makes x2 5x c a perfect
square trinomial. Write the expression as the square of
a binomial.
冢
冣
25
5 2
; x 4
2
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Your Notes
Solving a Quadratic Equation
Example 2
Solve the equation by completing the square.
2x2 8x 12 0
Solution
2x2 8x 12 0
x2 4x 6 0
x2 4x 6
x2 4x 22 6 4
(x 2)2 2
x 2 兹2
苶
x 2 i 兹2
苶
Write original equation.
Divide each side by the
coefficient of x 2.
Write the left side in the
form x 2 ⴙ bx.
Complete the square.
Write the left side as a
binomial squared.
Take square roots of
each side.
Solve for x.
Checkpoint Complete the following exercise.
2. Solve x2 4x 1 0 by completing the square.
2 兹3
苶
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Algebra 2 Notetaking Guide • Chapter 5
Your Notes
Example 3
Writing a Quadratic Function in Vertex Form
Write the quadratic function y x2 6x 7 in vertex form.
What is the vertex of the function’s graph?
y x2 6x 7
Original equation
y 9 (x2 6x 9 ) 7
Complete the square.
y 9 ( x 3 )2 7
Write as a binomial
squared.
y (x 3)2 2
Solve for y.
The vertex is at (3, 2) .
Checkpoint Complete the following exercises.
3. Write y x2 2x 4 in vertex form.
Homework
y (x 1)2 5
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