9-1: Complete The Square
Objectives:
1. To solve a quadratic
equation by finding a
square root
2. To solve quadratic
equations by
completing the
square
Assignment:
• P. 140: 10-13
• P. 149: 1-20
• Challenge Problems
Objective 1
Solving Quadratics
If a quadratic equation has no linear term,
you can use square roots to solve it.
•
By definition, if
x2 = c, then 𝑥 =
𝑐 and 𝑥 =
− 𝑐, usually
written 𝑥 = ± 𝑐
You would only solve a
quadratic by finding a
square root if it is of the
form
ax2 = c
Solving Quadratics
If a quadratic equation has no linear term,
you can use square roots to solve it.
By definition, if
x2 = c, then 𝑥 =
𝑐 and 𝑥 =
− 𝑐, usually
written 𝑥 = ± 𝑐
To solve a quadratic equation
using square roots:
1. Isolate
the
squared
term
2. Take the
square
root of
both
sides
Exercise 1a
Solve 2x2 – 15 = 35 for x.
Exercise 1b
Solve for x.
1
2
x

4

  11
3
Exercise 2
What must be true about a and c for ax2 = c
to have 2 roots, 1 root, or no roots?
Rationalizing the Denominator
We can use conjugates to get rid of radicals
in the denominator:
The process of multiplying the top and bottom
of a radical expression by the conjugate of
the denominator is called rationalizing the
denominator.
1
5

1 3 1

5 1 3
3

3
1 3 1
Fancy One



3


5  5 3
55 3

2
2
Exercise 3
Simplify the expression.
6
5
6
7 5
17
12
1
9 7
Exercise 4
Solve for x.
x2 – 10x + 25 = 35
Perfect Squares?
The previous exercise was an interesting
example, but not every trinomial is a
perfect square trinomial. However, we can
cleverly rearrange the terms to make any
trinomial into a perfect square.
This is called completing the square.
Objective 2
You will be
able to solve
quadratic functions
by completing the square
Algebra Tiles Activity!
In this activity, we will
be using algebra
tiles to learn how to
complete the
square. Don’t
worry; unlike most
algebra tile
activities, this one is
actually quite useful.
But first, what is an
algebra tile?
Geometric representation of
x
x2:
x
x2
x:
1
Unit Square:
x x
1
1 1
Algebra Tiles Activity!
1. You’ll need some tiles, so cut them out.
2. Use your tiles to represent x2 + 6x.
x2
x
x
x
x
x
x
3. What you want to accomplish is to make
your polynomial model into a square by
rearranging your tiles and adding the
correct number of unit squares.
To create this
x+3
square, put half of
your x’s on one
side of the square
x2
x x x
= (x + 3)2
and the other half
2
= x + 6x + 9
of your x’s on the
1 1 1
1 1 1
other side of the
1 1 1
square. Now fill in
the missing bits
with unit squares.
x x x
x+3
Algebra Tiles Activity!
Algebra Tiles Activity!
4. Now, let’s try again, this time
with the following expressions.
1
1
𝑥+1
2
4
4
𝑥+2
2
16
𝑥+4
2
25
𝑥+5
2
9
16
25
5. Based on
your
investigation,
how could
you complete
the square
given
x2 + bx?
half
In general then, to
x+3
complete the
square, you take
half of the middle
x2
x x x
= (x + 3)2
term and then
2
= x + 6x + 9
square it. The
1 1 1
1 1 1
number you get
1 1 1
3
becomes the
constant term.
x x x
x+3
Algebra Tiles Activity!
half
Notice also that the
x+3
number you get
from taking half of
the middle term
x2
x x x
= (x + 3)2
becomes the
2
= x + 6x + 9
constant of the
1 1 1
1 1 1
binomial that gets
1 1 1
3
squared.
x x x
x+3
Algebra Tiles Activity!
Algebra Tiles Activity!
x+2
= (x + 1)2
= x2 + 2x + 1
half
1
x x
= (x + 2)2
= x2 + 4x + 4
1 1
1 1
half
1
x2
x x
x
x+2
x2
x
x+1
x+1
2
Algebra Tiles Activity!
x+4
x x x x
= (x + 4)2
x x x x
= x2 + 8x + 16
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
half
x+4
x2
4
Algebra Tiles Activity!
x+5
x x x x x
= (x + 5)2
x x x x x
= x2 + 10x + 25
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
half
x+5
x2
5
Completing The Square
Exercise 5a
Find the value of c that makes x2 – 26x + c a
perfect square trinomial. The write the
expression as the square of a binomial.
Exercise 5b
Find the value of c that makes x2 + 7x + c a
perfect square trinomial. The write the
expression as the square of a binomial.
Exercise 5c
Find the value of c that the expression a
perfect square trinomial. The write the
expression as the square of a binomial.
1. x2 + 14x + c
2.
x2 – 22x + c
3.
x2 – 9x + c
Exercise 6
Solve by finding square roots.
1. x2 + 6x + 9 = 36
2.
x2 – 10x + 25 = 1
Solving Quadratics
Completing the square
allows you to solve
almost any quadratic
equation, regardless
of whether it is a
perfect square. This
is basically an
application of the
Addition Property of
Equality.
x2  8x  6  0
x 2  8 x ___  6
x 2  8 x  16  6  16
 x  4
2
 22
x  4   22
x  4 
22
Solving Quadratics
Solving a Quadratic Equation by
Completing the Square:
1. Use addition/subtraction to write your
equation in the form x2 + bx = c.
2. Complete the square on the quadratic
side of the equation and add this number
to both sides of the equation.
3. Factor and take the square root.
Exercise 7a
Solve 3x2 – 36x +150 = 0 by completing the
square.
Exercise 7b
Solve x2 – 9x + 20 = 0 by completing the
square.
Exercise 7c
Solve 2x2 – 12x + 7 = 0 by completing the
square.
9-1: Complete The Square
Objectives:
1. To solve a quadratic
equation by finding a
square root
2. To solve quadratic
equations by
completing the
square
Assignment:
• P. 140: 10-13
• P. 149: 1-20
• Challenge Problems