4.7
Complete the Square
Goal
Your Notes
p Solve quadratic equations by completing the
square.
VOCABULARY
Completing the square
COMPLETING THE SQUARE
Words To complete the square for the expression
Algebra
Example 1
1
2
2.
2
x 2 1 bx 1 1
2 5 1 x 1 }2b 21 x 1 }2b 2
2
51
2
x 2 1 bx, add
Make a perfect square trinomial
Find the value of c that makes x 2 1 12x 1 c a perfect
square trinomial. Then write the expression as the
square of a binomial.
Solution
1. Find half the coefficient of x.
2
5
5
2. Square the result of Step 1.
3. Replace c with the result of Step 2.
The trinomial x 2 1 12x 1 c is a perfect square when
c5
. Then
5(
108
Lesson 4.7 • Algebra 2 Notetaking Guide
)(
)5(
)2.
Copyright © Holt McDougal. All rights reserved.
4.7
Complete the Square
Goal
Your Notes
p Solve quadratic equations by completing the
square.
VOCABULARY
Completing the square The process that allows you
to write an expression of the form x 2 1 bx as the
square of a binomial
COMPLETING THE SQUARE
Words To complete the square for the expression
x 2 1 bx, add
1
b 2
.
2
}
2
Algebra x 2 1 bx 1
1
2
b 2
b
b
}
5 x1}
x
1
2
2
2
b 2
5 x1}
2
}
1
1
Example 1
21
2
2
Make a perfect square trinomial
Find the value of c that makes x 2 1 12x 1 c a perfect
square trinomial. Then write the expression as the
square of a binomial.
Solution
1. Find half the coefficient of x.
12
2
2. Square the result of Step 1.
6 2 5 36
3. Replace c with the result of Step 2.
x 2 1 12x 1 36
5 6
The trinomial x 2 1 12x 1 c is a perfect square when
c 5 36 . Then
x 2 1 12x 1 36 5 ( x 1 6 )( x 1 6 ) 5 ( x 1 6 )2.
108
Lesson 4.7 • Algebra 2 Notetaking Guide
Copyright © Holt McDougal. All rights reserved.
Your Notes
Checkpoint Find the value of c that makes the
expression a perfect square trinomial. Then write
the expression as the square of a binomial.
1. x 2 2 24x 1 c
Example 2
2. x 2 1 10x 1 c
Solve ax 2 1 bx 1 c 5 0 when a 5 1
Solve x 2 2 10x 1 13 5 0 by completing the square.
Solution
x 2 2 10x 1 13 5 0
5
Write left side in the
form x 2 1 bx.
5
Complete the square.
5
Write left side as a
binomial squared.
5
Take square roots of
each side.
x5
Solve for x.
x5
Simplify.
The solutions are
Copyright © Holt McDougal. All rights reserved.
Write original equation.
and
.
Lesson 4.7 • Algebra 2 Notetaking Guide
109
Your Notes
Checkpoint Find the value of c that makes the
expression a perfect square trinomial. Then write
the expression as the square of a binomial.
1. x 2 2 24x 1 c
2. x 2 1 10x 1 c
c 5 144
c 5 25
(x 2 12)2
(x 1 5)2
Example 2
Solve ax 2 1 bx 1 c 5 0 when a 5 1
Solve x 2 2 10x 1 13 5 0 by completing the square.
Solution
x 2 2 10x 1 13 5 0
Write original equation.
x 2 2 10x 5 213
Write left side in the
form x 2 1 bx.
x 2 2 10x 1 25 5 213 1 25
Complete the square.
(x 2 5)2 5 12
Write left side as a
binomial squared.
}
x 2 5 5 6Ï 12
Take square roots of
each side.
}
x 5 5 6 Ï12
Solve for x.
x 5 5 6 2Ï3
Simplify.
}
}
}
The solutions are 5 1 2Ï 3 and 5 2 2Ï3 .
Copyright © Holt McDougal. All rights reserved.
Lesson 4.7 • Algebra 2 Notetaking Guide
109
Your Notes
Until you learned
about complex
numbers in Lesson
4.6, you solved
equations over
the set of real
numbers. As you
continue through
the course, you will
include complex
solutions when you
solve equations.
Example 3
Solve ax 2 1 bx 1 c 5 0 when a Þ 1
Solve 3x 2 2 12x 1 27 5 0 by completing the square.
Solution
3x 2 2 12x 1 27 5 0
Write original equation.
5
Divide each side by the
coefficient of x 2.
5
Write left side in the form
x 2 1 bx.
5
Complete the square.
5
Write left side as binomial
squared.
5
Take square roots of each
side.
x5
The solutions are
Write in terms of the
imaginary unit i.
and
.
Checkpoint Solve the equations by completing the
square.
3. x 2 2 8x 1 7 5 0
110
Lesson 4.7 • Algebra 2 Notetaking Guide
4. 2x 2 2 20x 1 24 5 0
Copyright © Holt McDougal. All rights reserved.
Your Notes
Until you learned
about complex
numbers in Lesson
4.6, you solved
equations over
the set of real
numbers. As you
continue through
the course, you will
include complex
solutions when you
solve equations.
Example 3
Solve ax 2 1 bx 1 c 5 0 when a Þ 1
Solve 3x 2 2 12x 1 27 5 0 by completing the square.
Solution
3x 2 2 12x 1 27 5 0
Write original equation.
x 2 2 4x 1 9 5 0
Divide each side by the
coefficient of x 2.
x 2 2 4x 5 29
Write left side in the form
x 2 1 bx.
x 2 2 4x 1 4 5 29 1 4
(x 2 2)2 5 25
Complete the square.
Write left side as binomial
squared.
}
x 2 2 5 6Ï 25
Take square roots of each
side.
}
x 5 2 6 i Ï5
Write in terms of the
imaginary unit i.
}
}
The solutions are 2 1 i Ï 5 and 2 2 i Ï5 .
Checkpoint Solve the equations by completing the
square.
3. x 2 2 8x 1 7 5 0
1, 7
110
Lesson 4.7 • Algebra 2 Notetaking Guide
4. 2x 2 2 20x 1 24 5 0
}
5 6 Ï 13
Copyright © Holt McDougal. All rights reserved.
Your Notes
Write a quadratic function in vertex form
Example 4
Write y 5 x 2 1 14x 1 44 in vertex form. Then identify
the vertex.
Solution
y 5 x 2 1 14x 1 44
y1
5 (x 2 1 14x 1
y1
5(
Write original
function.
) 1 44
)2 1 44
Complete the
square.
Write as a binomial
squared.
y5
Solve for y.
The vertex form of the function is y 5
The vertex is (
,
).
.
Checkpoint Write the quadratic function in vertex
form. Then identify the vertex.
5. y 5 x 2 2 12x 1 22
6. y 5 x 2 1 16x 1 53
Homework
Copyright © Holt McDougal. All rights reserved.
Lesson 4.7 • Algebra 2 Notetaking Guide
111
Your Notes
Example 4
Write a quadratic function in vertex form
Write y 5 x 2 1 14x 1 44 in vertex form. Then identify
the vertex.
Solution
y 5 x 2 1 14x 1 44
Write original
function.
y 1 49 5 (x 2 1 14x 1 49 ) 1 44
Complete the
square.
y 1 49 5 ( x 1 7 )2 1 44
Write as a binomial
squared.
y 5 (x 1 7)2 2 5
Solve for y.
The vertex form of the function is y 5 (x 1 7)2 2 5 .
The vertex is ( 27 , 25 ).
Checkpoint Write the quadratic function in vertex
form. Then identify the vertex.
5. y 5 x 2 2 12x 1 22
6. y 5 x 2 1 16x 1 53
y 5 (x 2 6)2 2 14
y 5 (x 1 8)2 2 11
vertex: (6, 214)
vertex: (28, 211)
Homework
Copyright © Holt McDougal. All rights reserved.
Lesson 4.7 • Algebra 2 Notetaking Guide
111