9-3B Completing the Square

Adapted from Math 8H Completing the Square JoAnn Evans

(to solve quadratic equations)

You will need a calculator for today’s lesson.

Add a few problems where the middle term is odd!

07x6x factoringbySolve 2

prime

First, a few review topics to refresh your memory………

Solving a quadratic by taking the square root:

To solve for x, take the square root of both sides of the equation.

When taking the square root of both sides of the equation, you must add a plus/minus sign because 81 has both a positive and a negative root.

81x2

81x2

9x

Another look at the characteristics common to all

PERFECT SQUARE TRINOMIALS :

x2 + 6x + 9

x2 - 12x + 36

x2 + 2x + 1

x2 + x + ¼

In each trinomial, take a look at the coefficient of the middle term and the constant. What is the relationship between these two numbers?

Take half of the middle term coefficient.

If you square that number, you will

have the constant in each case.

Why is that? Think of the factored form of each P.S.T.

(x + 3)2

(x – 6)2

(x + 1)2

(x + ½) 2

What is the missing constant that will complete the square?

x4x 2

2x 2

4

x6x2

What is the missing constant that will complete the square?

23x

9

x6x2

What is the missing constant that will complete the square?

23x

9

x8x2

What is the missing constant that will complete the square?

24x

16

Make each incomplete trinomial into a PERFECT SQUARE TRINOMIAL by finding the missing constant:

x2 + 8x + ___

x2 – 14x + ___

x2 + 10x + ___

x2 + 22x + ___

x2 - 20x + ___

16

49

25

121

100

Factored form: (x + 4)2

Factored form: (x - 7)2

Factored form: (x + 5)2

Factored form: (x + 11)2

Factored form: (x - 10)2

To solve a quadratic by “Completing the Square”, follow these steps:

1. Make sure the coefficient of x2 is 1.

2. Move everything to the LEFT side of the equation EXCEPT the constant.

3. Make the left hand side of the equation into a PERFECT SQUARE TRINOMIAL by following the steps we just practiced.

4. Remember, if you add a number to one side of an equation, you must add the same number to the other side of the equation.

5. Factor the left side into the SQUARE OF A BINOMIAL.

6. Take the square root of each side. Remember to add the ± symbol to the right side.

7. Solve for x.

(x – 3)2 = 2

(half of -6)2

Solve by completing the square.x2 – 6x + 7 = 0

x2 – 6x + ___ = -7

x2 – 6x + 9 = -7 + 9

x2 – 6x + 9 = 2 Simplify.

2)3x( 2 Find the square root of each side. Solve for x.

23x 23x

Add 9 to both sides.

Keep everything except the constant on the left side. Leave a space for the number that will make a

perfect square trinomial. What’s half of -6, squared?

6.1,4.4

Rewrite as a binomial squared.

Example 1 Solve by completing the square. Round to the nearest tenth if necessary.

x2 – 4x + 2 = 0 Keep everything except the constant on the left side.

x2 – 4x + ___ = -2 Leave a space for the number that will make a perfect

square trinomial. What’s half of -4, squared? x2 – 4x + 4 = -2 +

4

x2 – 4x + 4 = 2 Simplify.

(half of -4)2

(x – 2)2 = 2 Rewrite as a binomial squared.

2)2x( 2 Find the square root of each

side.Solve for x.

22x 22x

Add 4 to both sides.

6.0,4.3

Example 2 Solve by completing the square. Round to nearest tenth if necessary.

x2 - 12x + 4 = 0 Keep everything except the constant on the left side. x2 - 12x + ___ = -4 Leave a space for the number that will make a

perfect square trinomial. What’s half of -12, squared?

x2 - 12x + 36 = -4 + 36

x2 - 12x + 36 = 32 Simplify.

(half of -12)2

(x - 6)2 = 32 Rewrite as a binomial squared.

Find the square root of each side. Solve for x.

Add 36 to both sides.

32)6x( 2

326x

326x 4.0,7.11

Solve by completing the square. Round to the nearest tenth if necessary.

x2 + 6x - 8 = 0

Example 3

Example 4 x2 - 2x - 3 = 0

Example 5 x2 + 16x + 4 = 0

Example 6 x2 + 10x - 24 = 0

Example 3 Solve by completing the square. Round to the nearest tenth if necessary

x2 + 6x - 8 = 0 Keep everything except the constant on the left side.

x2 + 6x + ___ = 8 Leave a space for the number that will make a

perfect square trinomial. What’s half of 6, squared?

x2 + 6x + 9 = 8 + 9

x2 + 6x + 9 = 17 Simplify.

(half of 6)2

(x + 3)2 = 17 Rewrite as a binomial squared.

Find the square root of each side. Solve for x.

Add 9 to both sides.

17)3x( 2

173x

173x 1.7,1.1

Example 4 Solve by completing the square. Round to the nearest tenth if necessary.

x2 - 2x - 3 = 0 Keep everything except the constant on the left side.

x2 - 2x + ___ = 3 Leave a space for the number that will make a

perfect square trinomial. What’s half of -2, squared? x2 - 2x + 1 = 3 + 1

x2 - 2x + 1 = 4 Simplify.

(half of -2)2

(x - 1)2 = 4 Rewrite as a binomial squared.

Find the square root of each side. Solve for x.

Add 1 to both sides.

4)1x( 2

21x

1,3 21x

Example 5 Solve by completing the square. Round to the nearest tenth if necessary.

x2 + 16x + 4 = 0 Keep everything except the constant on the left side. x2 + 16x + ___ = -4 Leave a space for the number that will make a

perfect square trinomial. What’s half of 16, squared? x2 + 16x + 64 = -4 + 64

x2 + 16x + 64 = 60 Simplify.

(half of 16)2

(x + 8)2 = 60 Rewrite as a binomial squared.

Find the square root of each side. Solve for x.

Add 64 to both sides.

60)8x( 2

608x 1528x

7.15,3.0

Example 6 Solve by completing the square. Round to the nearest tenth if necessary.

x2 + 10x - 24 = 0 Keep everything except the constant on the left side. x2 + 10x + ___ = 24 Leave a space for the number that will make a

perfect square trinomial. What’s half of 5, squared? x2 + 10x + 25 = 24 + 25

x2 + 10x + 25 = 49 Simplify.

(half of 10)2

(x + 5)2 = 49 Rewrite as a binomial squared.

Find the square root of each side. Solve for x.

Add 25 to both sides.

49)5x( 2

75x

12,2 75x

Solve by completing the square.

x2 + 6x - 8 = 0

Example 3

Example 4 x2 - 2x - 3 = 0

Example 5 x2 + 16x + 4 = 0

Example 6 x2 + 10x - 24 = 0

1,3

1.7,1.1

7.15,3.0

12,2

9-A9 Page 489-491, # 16–27, 48-52, 62-65.