Do Now:

• Pass out calculators. • Write down your assignments for the week for a scholar

dollar. • Use square roots to solve the quadratic equations below: 1. 3x2=272. P2 + 12 = 123. a2 +3 = -4 4. 4(x +6)2 = 32 5. 20 = 2(m +5)2

Answers:

1. -3, 3

2. 0

3. No solution

4. -3.17, -8.83

5. -8.16, -1.84

Objective:

• To solve quadratic equations by completing the square.

Completing the Square:

• For an expression of the form: x2 + bx, you can add a constant c to the expression so that the expression x2 + bx + c is a perfect square trinomial.

*This process is called completing the square.

Completing the Square:

• To complete the square for the expression x2 + bx ..

Add the square root of half the coefficient of the term bx.

So.. x2 + bx + (b/2)2 = (x + b/2)2

EXAMPLE 1 Complete the square

Find the value of c that makes the expression x2 + 5x + c a perfect square trinomial. Then write the expression as the square of a binomial.

STEP 1Find the value of c. For the expression to be a perfect square trinomial, c needs to be the square of half the coefficient of bx.

Find the square of half the coefficient of bx.2

2 = 254c = 5

EXAMPLE 1 Complete the square

STEP 2Write the expression as a perfect square trinomial. Then write the expression as the square of a binomial.

Substitute 25 for c.

4

x2 + 5x + c = x2 + 5x +254

Square of a binomial52

2+x =

GUIDED PRACTICE for Example 1

Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial.

1. x2 + 8x + c ANSWER 16; (x + 4)2

2. x2 12x + c

3. x2 + 3x + c

ANSWER 36; (x 6)2

ANSWER ; (x )294

32

EXAMPLE 2 Solve a quadratic equation

Solve x2 – 16x = –15 by completing the square.

SOLUTION

Write original equation.x2 – 16x = –15

Add , or (– 8)2, to each side.

– 16 2

2

x2 – 16x + (– 8)2 = –15 + (– 8)2

Write left side as the square of a binomial.

(x – 8)2 = –15 + (– 8)2

Simplify the right side.(x – 8)2 = 49

EXAMPLE 2 Standardized Test Practice

Take square roots of each side.x – 8 = ±7

Add 8 to each side.x = 8 ± 7

ANSWER

The solutions of the equation are 8 + 7 = 15 and 8 – 7 = 1.

EXAMPLE 2 Standardized Test Practice

CHECK

You can check the solutions in the original equation.

If x = 15:

(15)2 – 16(15) –15 ?=

–15 = –15

If x = 1:

(1)2 – 16(1) –15 ?=

–15 = –15

EXAMPLE 3 Solve a quadratic equation in standard form

Solve 2x2 + 20x – 8 = 0 by completing the square.

SOLUTION

Write original equation.2x2 + 20x – 8 = 0

Add 8 to each side.2x2 + 20x = 8

Divide each side by 2.x2 + 10x = 4

Add 10 2

2, or 52, to each side.x2 + 10x + 52 = 4 + 52

Write left side as the square of a binomial.

(x + 5)2 = 29

EXAMPLE 3 Solve a quadratic equation in standard form

Take square roots of each side.x + 5 =

± 29

Subtract 5 from each side.x = –5 ± 29

ANSWER

The solutions are – 5 + 29 0.39 and – 5 – 29 –10.39.

GUIDED PRACTICE for Examples 2 and 3

4. x2 – 2x = 3

ANSWER 1, 3

5. m2 + 10m = –8

ANSWER 9.12, 0.88

6. 3g2 – 24g + 27 = 0

ANSWER 1.35, 6.65

Exit Ticket:Find the value of c that makes the expression a perfect square trinomial. Then write the expression as the square of a binomial. 1.x2 - 3x + c

Complete the square.2. x2 + 10x = 24 3. 2x2 + 24x + 10 = 0

Answers: 1. 9/4; (x - 3/2)2 2. -12, 2 3. -11.57, -0.43

EXAMPLE 4 Solve a multi-step problem

CRAFTS

You decide to use chalkboard paint to create a chalkboard on a door. You want the chalkboard to have a uniform border as shown. You have enough chalkboard paint to cover 6 square feet. Find the width of the border to the nearest inch.

EXAMPLE 4 Solve a multi-step problem

SOLUTION

STEP 1

Write a verbal model. Then write an equation. Let x be the width (in feet) of the border.

6 = (7 – 2x) (3 – 2x)

EXAMPLE 4 Solve a multi-step problem

STEP 2

Solve the equation.

Write equation.6 = (7 – 2x)(3 – 2x)

Multiply binomials.6 = 21 – 20x + 4x2

Subtract 21 from each side.–15 = 4x2 – 20x

Divide each side by 4. – = x2 – 5x154

Add –52

, or 25 4

, to each side.225 4

– 15 4

+ = x2 – 5x + 25 4

EXAMPLE 4 Solve a multi-step problem

Write right side as the square of a binomial.

– 15 4 +

25 4 = (x – )25

2

Simplify left side.52 = (x – )5

22

Take square roots of each side. ± 52

= x – 52

Add52 to each side.

52 ± 5

2= x

EXAMPLE 4 Solve a multi-step problem

The solutions of the equation are

and It is not possible for the width

of the border to be 4.08 feet because the width of the door is 3 feet. So, the width of the border is 0.92 foot. Convert 0.92 foot to inches.

52

52

+ 4.08

52

52

0.92

0.92 ft12 in.

1 ft= 11.04 in Multiply by conversion factor

ANSWER

The width of the border should be about 11 inches.