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
HOMEWORK
READ textbook section 2-4, pages 85 - 88
TAKE NOTES
- vocabulary
- “Know it! note”
- Reading Math
- “Caution!”
- Helpful Hint
BE READY to answer questions from the reading and
note taking
****** READ but DO NOT copy the examples
Holt McDougal Algebra 2
2-4
Completing the Square
METHODS  solving quadratic equations
1.Making a table of values and GRAPHING
•
vertex
•
axis of symmetry (reflection points)
•
y-intercept
•
concavity
Holt McDougal Algebra 2
2-4
Completing the Square
METHODS  solving quadratic equations
2.FACTORING
•
GCF
•
leading coefficient a = 1 and a  1
•
difference of squares
•
perfect square trinomials
3.Using the SQUARE ROOT PROPERTY
4.COMPLETING THE SQUARE
Holt McDougal Algebra 2
2-4
Completing the Square
SQUARE ROOT PROPERTY
• take the square root of both sides of an equation
• REMEMBER: BOTH positive & negative roots
For example:
𝟐
𝒙 =𝒂
𝒙𝟐 = ± 𝒂
𝒙 = 𝒂 or 𝒙 = − 𝒂
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
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
Check
Use a graphing calculator.
Check BOTH SOLUTIONS (POS. & NEG)
Holt McDougal Algebra 2
2-4
Completing the Square
Example 1B: Solving Equations by Using the Square
Root Property
Solve the equation.
(x + 6)2 = 28
x2 + 12x + 36 = 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
YOU TRY
Solve the equation.
4x2 = 25
25
𝒙 =
4
𝟐
4x2 – 20 = 5
Add 20 to 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
Check
Use a graphing calculator.
Holt McDougal Algebra 2
2-4
Completing the Square
YOU TRY
Solve the equation.
(x + 4)2 = 49
x2 + 8x + 16 = 49
Factor the perfect square trinomial.
Take the square root of both sides.
x = –4 ± 49
Subtract 4 from both sides.
x = –11, 3
Simplify.
Holt McDougal Algebra 2
2-4
Completing the Square
Check
Use a graphing calculator.
Holt McDougal Algebra 2
2-4
Completing the Square
COMPLETING THE SQUARE
We have looked at expressions that were
perfect squares.
 If NOT given a perfect square trinomial,
  use algebra to REWRITE the quadratic
expression as a perfect square
Holt McDougal Algebra 2
2-4
Completing the Square
Completing the square
USE  x2 + bx of the quadratic expression
 to make a perfect square trinomial
 To complete the square of
x2 + bx, add
.
Holt McDougal Algebra 2
𝒃 𝟐
𝟐
2-4
Completing the Square
Example: 2A
Complete the square for the expression.
Write the resulting expression as a
binomial squared
x2 – 14 +
Holt McDougal Algebra 2
2-4
Completing the Square
x2 – 14 +
b = – 14
14
−
2
−7
2
2
divide it by 2 and square it
= 49
𝑥 − 14 + 𝟒𝟗
𝑥−7
identify b
2
Holt McDougal Algebra 2
simplify
add to quadratic expression
factor the quadratic expression
2-4
Completing the Square
Check
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
.
2-4
Completing the Square
Check
Find the square of the binomial.
Holt McDougal Algebra 2
2-4
Completing the Square
YOU TRY
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 + 4x +
Find
x2 + 4x + 4
(x + 2)2
Check
.
Add.
Factor.
Find the square of the binomial.
(x + 2)2 = (x + 2)(x + 2)
= x2 + 4x + 4
Holt McDougal Algebra 2
2-4
Completing the Square
YOU TRY
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 – 4x +
Find
x2 – 4x + 4
(x – 2)2
Check
.
Add.
Factor.
Find the square of the binomial.
(x – 2)2 = (x – 2)(x – 2)
= x2 – 4x + 4
Holt McDougal Algebra 2
2-4
Completing the Square
YOU TRY
Complete the square for the expression. Write
the resulting expression as a binomial squared.
x2 + 3x +
Find
Add.
Factor.
Holt McDougal Algebra 2
.
Check Find the square
of the binomial.
2-4
Completing the Square
Solve a Quadratic Equation by
 COMPLETING THE SQUARE
STEPS:
Holt McDougal Algebra 2
2-4
Completing the Square
STEPS:
1. Collect variable terms on one side of the
equation and constants on the other.
2. As needed, divide both sides by a to make the
coefficient of the 𝒙𝟐 -term 1.
3. Complete the square by adding
sides of the equation.
Holt McDougal Algebra 2
𝑏 2
2
to both
2-4
Completing the Square
STEPS:
4. Factor the variable expression as a
perfect square.
5. Take the square root of both sides of the
equation.
6. Solve for the values of the variable.
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
YOU TRY
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
YOU TRY
Solve the equation by completing the square.
3x2 – 24x = 27
x2 – 8x = 9
x2 –8x +
Divide both sides by 3.
=9+
Set up to complete the
square.
Add
to both sides.
Simplify.
Holt McDougal Algebra 2
2-4
Completing the Square
Check It Out! Example 3b Continued
Solve the equation by completing the square.
Factor.
Take the square root
of both sides.
Simplify.
x – 4 =–5 or x – 4 = 5
x =–1 or x = 9
Holt McDougal Algebra 2
Solve for x.
2-4
Completing the Square
HOMEWORK
Textbook pages 89 – 90
#’s 3-13, 21-31,41-49, 51-59
ODD problems ONLY
Holt McDougal Algebra 2