2-4
2-4 Completing
Completing the
the Square
Square
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Holt
McDougal
Algebra 2Algebra
Algebra22
Holt
McDougal
2-4
Completing the Square
Warm Up
Write each expression as a trinomial.
1. (x – 5)2
x2 – 10x + 25
2. (3x + 5)2
9x2 + 30x + 25
Factor each expression.
3. x2 – 18x + 81 (x – 9)2
4. 16x2 + 24x + 9 (4x + 3)2
Holt McDougal Algebra 2
2-4
Completing the Square
Objectives
Solve quadratic equations by
completing the square.
Write quadratic equations in vertex
form.
Holt McDougal Algebra 2
2-4
Completing the Square
Vocabulary
completing the square
Holt McDougal Algebra 2
2-4
Completing the Square
Essential Question
• How do you find the term needed to
complete the square?
Holt McDougal Algebra 2
2-4
Completing the Square
Many quadratic equations contain expressions
that cannot be easily factored. For equations
containing these types of expressions, you can
use square roots to find roots.
The most efficient method to solve a quadratic
equation using the square root method is when
the linear term is missing or when the trinomial is
a perfect square.
Holt McDougal Algebra 2
2-4
Completing the Square
Reading Math
Read
as “plus or minus square root of a.”
Holt McDougal Algebra 2
2-4
Completing the Square
Properties of Square Roots
(a>0, b>0)
• Product Property
• Example:
18 = 9 ∙ 2 = 9 ∙ 2 = 3 2
• Quotient Property
• Example:
Holt McDougal Algebra 2
𝑎𝑏 = 𝑎 ∙ 𝑏
2

25
a

b
2
2

5
25
a
b
2-4
Completing the Square
Example 1A: Solving Equations by Using the Square
Root Property
Solve the equation.
4x2 + 11 = 59
4x2 = 48
x2 = 12
Subtract 11 from both sides.
Divide both sides by 4 to isolate the
square term.
Take the square root of both sides.
Simplify.
Holt McDougal Algebra 2
2-4
Completing the Square
Example 1A Continued
Check Use a graphing calculator and substitute
the roots for the variable.
Holt McDougal Algebra 2
2-4
Completing the Square
Example 1B: Solving Equations by Using the Square
Root Property
Solve the equation.
x2 + 12x + 36 = 28
(x + 6)2 = 28
Factor the perfect square trinomial
Take the square root of both sides.
Subtract 6 from both sides.
Simplify.
Holt McDougal Algebra 2
2-4
Completing the Square
Example 1B Continued
Check
Use a graphing calculator.
Holt McDougal Algebra 2
2-4
Completing the Square
If a quadratic expression of the form x2 + bx
cannot model a square, you can add a term to
form a perfect square trinomial. This is called
completing the square.
Holt McDougal Algebra 2
2-4
Completing the Square
Example 2A: Completing the Square
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 – 14x +
Find
x2 – 14x + 49
(x – 7)2
Check
.
Add to the expression.
Factor.
Find the square of the binomial.
(x – 7)2 = (x – 7)(x – 7)
= x2 – 14x + 49
Holt McDougal Algebra 2
2-4
Completing the Square
Example 2B: Completing the Square
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 + 9x +
Find
Add.
Factor.
Holt McDougal Algebra 2
.
Check Find the square
of the binomial.
2-4
Completing the Square
You can complete the square to solve quadratic
equations.
Holt McDougal Algebra 2
2-4
Completing the Square
Example 3A: Solving a Quadratic Equation by
Completing the Square
Solve the equation by completing the square.
x2 = 12x – 20
x2
– 12x = –20
x2 – 12x +
= –20 +
Collect variable terms on
one side.
Set up to complete the
square.
Add
x2 – 12x + 36 = –20 + 36
Holt McDougal Algebra 2
Simplify.
to both sides.
2-4
Completing the Square
Example 3A Continued
(x – 6)2 = 16
Factor.
Take the square root of
both sides.
x – 6 = ±4
x – 6 = 4 or x – 6 = –4
x = 10 or x = 2
Holt McDougal Algebra 2
Simplify.
Solve for x.
2-4
Completing the Square
Example 3B: Solving a Quadratic Equation by
Completing the Square
Solve the equation by completing the square.
18x + 3x2 = 45
x2 + 6x = 15
x2 + 6x +
= 15 +
Divide both sides by 3.
Set up to complete the
square.
Add
x2 + 6x + 9 = 15 + 9
Holt McDougal Algebra 2
to both sides.
Simplify.
2-4
Completing the Square
Example 3B Continued
(x + 3)2 = 24
Factor.
Take the square root of
both sides.
Simplify.
Holt McDougal Algebra 2
2-4
Completing the Square
Check It Out! Example 3a
Solve the equation by completing the square.
x2 – 2 = 9x
Collect variable terms on
one side.
x2 – 9x = 2
x2 – 9x +
=2+
Set up to complete the
square.
Add
to both sides.
Simplify.
Holt McDougal Algebra 2
2-4
Completing the Square
Check It Out! Example 3a Continued
Factor.
9
x –  ± 89
4
2
x
Holt McDougal Algebra 2
9 ± 89
2
Take the square root of
both sides.
Simplify.
2-4
Completing the Square
Recall the vertex form of a quadratic function
from lesson 5-1: f(x) = a(x – h)2 + k, where the
vertex is (h, k).
You can complete the square to rewrite any
quadratic function in vertex form.
Helpful Hint
In Example 3, the equation was balanced by
adding
to both sides. Here, the function is
balanced by adding and subtracting
side.
Holt McDougal Algebra 2
on one
2-4
Completing the Square
Example 4A: Writing a Quadratic Function in Vertex
Form
Write the function in vertex form, and identify
its vertex.
f(x) = x2 + 16x – 12
f(x)=(x2
+ 16x +
) – 12 –
Set up to complete
the square.
Add and subtract
f(x) = (x + 8)2 – 76
Simplify and factor.
Because h = –8 and k = –76, the vertex is (–8, –76).
Holt McDougal Algebra 2
.
2-4
Completing the Square
Example 4A Continued
Check Use the axis of symmetry formula to
confirm vertex.
y = f(–8) = (–8)2 + 16(–8) – 12 = –76 
Holt McDougal Algebra 2
2-4
Completing the Square
Example 4B: Writing a Quadratic Function in Vertex
Form
Write the function in vertex form, and identify
its vertex
g(x) = 3x2 – 18x + 7
g(x) = 3(x2 – 6x) + 7
g(x) = 3(x2 – 6x +
)+7–
2
Factor so the coefficient
of x2 is 1.
Set up to complete the
square.
Add
. Because
is multiplied by 3, you
must subtract 3
.
Holt McDougal Algebra 2
2-4
Completing the Square
Example 4B Continued
g(x) = 3(x – 3)2 – 20
Simplify and factor.
Because h = 3 and k = –20, the vertex is (3, –20).
Check A graph of the
function on a
graphing calculator
supports your
answer.
Holt McDougal Algebra 2
2-4
Completing the Square
Check It Out! Example 4a
Write the function in vertex form, and identify
its vertex
f(x) = x2 + 24x + 145
f(x) = (x2 + 24x +
) + 145 –
Set up to complete
the square.
Add and subtract
f(x) = (x + 12)2 + 1
Simplify and factor.
Because h = –12 and k = 1, the vertex is (–12, 1).
Holt McDougal Algebra 2
.
2-4
Completing the Square
Check It Out! Example 4a Continued
Check Use the axis of symmetry formula to
confirm vertex.
y = f(–12) = (–12)2 + 24(–12) + 145 = 1 
Holt McDougal Algebra 2
2-4
Completing the Square
Check It Out! Example 4b
Write the function in vertex form, and identify
its vertex
g(x) = 5x2 – 50x + 128
g(x) = 5(x2 – 10x) + 128
g(x) = 5(x2 – 10x +
) + 128 –
Factor so the coefficient
of x2 is 1.
Set up to complete the
square.
Add
. Because
is multiplied by 5, you
must subtract 5
.
Holt McDougal Algebra 2
2-4
Completing the Square
Check It Out! Example 4b Continued
g(x) = 5(x – 5)2 + 3
Simplify and factor.
Because h = 5 and k = 3, the vertex is (5, 3).
Check A graph of the
function on a
graphing calculator
supports your
answer.
Holt McDougal Algebra 2
2-4
Completing the Square
Lesson Quiz
1. Complete the square for the expression
x2 – 15x +
. Write the resulting expression
as a binomial squared.
Solve each equation.
2. x2 – 16x + 64 = 20
3. x2 – 27 = 4x
Write each function in vertex form and
identify its vertex.
5. f(x) = 2x2 – 12x – 27
4. f(x)= x2 + 6x – 7
f(x) = (x + 3)2 – 16;
(–3, –16)
Holt McDougal Algebra 2
f(x) = 2(x – 3)2 – 45;
(3, –45)
2-4
Completing the Square
Essential Question
• How do you find the term needed to
complete the square?
• Find half of the coefficient of the linear
term (bx) and square it.
𝑏 2
2
• The trinomial will be a perfect square and
will factor into 𝑥 ±
Holt McDougal Algebra 2
𝑏 2
2
2-4
Completing the Square
• How does a ghost solve a quadratic
equation?
• By completing the scare.
Holt McDougal Algebra 2