Holt McDougal Algebra 1

8-8 Completing the Square8-8 Completing the Square

Holt Algebra 1

Warm Up

Lesson Presentation

Lesson Quiz

Holt McDougal Algebra 1

Holt McDougal Algebra 1

8-8 Completing the Square

Warm Up

Simplify.

19 1. 2.

3. 4.

Holt McDougal Algebra 1

8-8 Completing the Square

Warm Up

Solve each quadratic equation by factoring.

5. x2 + 8x + 16 = 0

6. x2 – 22x + 121 = 0

7. x2 – 12x + 36 = 0

x = –4

x = 11

x = 6

Holt McDougal Algebra 1

8-8 Completing the Square

Solve quadratic equations by completing the square.

Objective

Holt McDougal Algebra 1

8-8 Completing the Square

completing the square

Vocabulary

Holt McDougal Algebra 1

8-8 Completing the Square

In the previous lesson, you solved quadratic equations by isolating x2 and then using square roots. This method works if the quadratic equation, when written in standard form, is a perfect square.

When a trinomial is a perfect square, there is a relationship between the coefficient of the x-term and the constant term.

X2 + 6x + 9 x2 – 8x + 16 Divide the coefficient of the x-term by 2, then

square the result to get

the constant term.

Holt McDougal Algebra 1

8-8 Completing the Square

An expression in the form x2 + bx is not a perfect square. However, you can use the relationship shown above to add a term to x2 + bx to form a trinomial that is a perfect square. This is called completing the square.

Holt McDougal Algebra 1

8-8 Completing the Square

Example 1: Completing the Square

Complete the square to form a perfect square trinomial.

A. x2 + 2x + B. x2 – 6x +

x2 + 2x

x2 + 2x + 1

x2 + –6x

x2 – 6x + 9

Identify b.

.

Holt McDougal Algebra 1

8-8 Completing the Square

Check It Out! Example 1

Complete the square to form a perfect square trinomial.

a. x2 + 12x + b. x2 – 5x +

x2 + 12x

x2 + 12x + 36

x2 + –5xIdentify b.

x2 – 5x +

.

Holt McDougal Algebra 1

8-8 Completing the Square

Check It Out! Example 1

Complete the square to form a perfect square trinomial.

c. 8x + x2 +

x2 + 8x

x2 + 8x + 16

Identify b.

.

Holt McDougal Algebra 1

8-8 Completing the Square

To solve a quadratic equation in the form x2 + bx = c, first complete the square of x2 + bx. Then you can solve using square roots.

Holt McDougal Algebra 1

8-8 Completing the Square

Solving a Quadratic Equation by Completing the Square

Holt McDougal Algebra 1

8-8 Completing the Square

Example 2A: Solving x2 +bx = c by Completing the Square

Solve by completing the square. Check your answer.

x2 + 16x = –15

Step 1 x2 + 16x = –15

Step 2

Step 3 x2 + 16x + 64 = –15 + 64

Step 4 (x + 8)2 = 49

Step 5 x + 8 = ± 7

Step 6 x + 8 = 7 or x + 8 = –7

x = –1 or x = –15

The equation is in the

form x2 + bx = c.

Complete the square.

Factor and simplify.

Take the square root

of both sides.

Write and solve two

equations.

.

Holt McDougal Algebra 1

8-8 Completing the Square

Example 2A Continued

Solve by completing the square. Check your answer.

x2 + 16x = –15

The solutions are –1 and –15.

Check x2 + 16x = –15

(–1)2 + 16(–1) –15

1 – 16 –15

–15 –15✓

x2 + 16x = –15

(–15)2 + 16(–15) –15

225 – 240 –15

–15 –15✓

Holt McDougal Algebra 1

8-8 Completing the Square

Example 2B: Solving x2 +bx = c

Solve by completing the square. Check your answer.

x2 – 4x – 6 = 0

Step 1 x2 + (–4x) = 6

Step 3 x2 – 4x + 4 = 6 + 4

Step 4 (x – 2)2 = 10

Step 5 x – 2 = ± √10

Write in the form

x2 + bx = c.

Complete the square.

Factor and simplify.

Take the square root

of both sides.

Write and solve two

equations.

Step 6 x – 2 = √10 or x – 2 = –√10

x = 2 + √10 or x = 2 – √10

.Step 2

Holt McDougal Algebra 1

8-8 Completing the Square

Example 2B Continued

Solve by completing the square. Check your answer.

The solutions are 2 + √10 and x = 2 – √10.

Check Use a graphing calculator to check your answer.

Holt McDougal Algebra 1

8-8 Completing the Square

Check It Out! Example 2a

Solve by completing the square. Check your answer.

x2 + 10x = –9

Step 1 x2 + 10x = –9

Step 3 x2 + 10x + 25 = –9 + 25 Complete the square.

The equation is in the

form x2 + bx = c.

Step 2

Step 4 (x + 5)2 = 16

Step 5 x + 5 = ± 4

Step 6 x + 5 = 4 or x + 5 = –4

x = –1 or x = –9

Factor and simplify.

Take the square root

of both sides.

Write and solve two

equations.

.

Holt McDougal Algebra 1

8-8 Completing the Square

Check It Out! Example 2a Continued

Solve by completing the square. Check your answer.

x2 + 10x = –9

The solutions are –9 and –1.

x2 + 10x = –9

(–9)2 + 10(–9) –9

81 – 90 –9

–9 –9 ✓

x2 + 10x = –9

(–1)2 + 10(–1) –9

1 – 10 –9

–9 –9 ✓

Check

Holt McDougal Algebra 1

8-8 Completing the Square

Check It Out! Example 2b

Solve by completing the square. Check your answer.

t2 – 8t – 5 = 0

Step 1 t2 + (–8t) = 5

Step 3 t2 – 8t + 16 = 5 + 16 Complete the square.

Write in the form

x2 + bx = c.

Step 2

Step 4 (t – 4)2 = 21

Step 5 t – 4 = ± √21

Factor and simplify.

Take the square root

of both sides.

Write and solve two

equations.

Step 6 t = 4 + √21 or t = 4 – √21

.

Holt McDougal Algebra 1

8-8 Completing the Square

Check It Out! Example 2b Continued

Solve by completing the square. Check your answer.

t = 4 – √21 or t = 4 + √21.The solutions are

Check Use a graphing calculator to check your answer.

Holt McDougal Algebra 1

8-8 Completing the Square

Example 3A: Solving ax2 + bx = c by Completing the Square

Solve by completing the square.

–3x2 + 12x – 15 = 0

Step 1

x2 – 4x + 5 = 0 x2 – 4x = –5

x2 + (–4x) = –5

Step 3 x2 – 4x + 4 = –5 + 4

Divide by – 3 to make a = 1.

Write in the form x2 + bx = c.

Complete the square.

.Step 2

Holt McDougal Algebra 1

8-8 Completing the Square

Example 3A Continued

Solve by completing the square.

–3x2 + 12x – 15 = 0

Step 4 (x – 2)2 = –1

There is no real number whose square is negative, so there are no real solutions.

Factor and simplify.

Holt McDougal Algebra 1

8-8 Completing the Square

Example 3B: Solving ax2 + bx = c by Completing the Square

Solve by completing the square.

5x2 + 19x = 4

Step 1 Divide by 5 to make a = 1.

Write in the form x2 + bx = c.

Step 2 .

Holt McDougal Algebra 1

8-8 Completing the Square

Complete the square.Step 3

Example 3B Continued

Solve by completing the square.

Factor and simplify.Step 4

Step 5 Take the square root

of both sides.

Rewrite using like

denominators.

Holt McDougal Algebra 1

8-8 Completing the Square

Example 3B Continued

Solve by completing the square.

Step 6

The solutions are and –4.

Write and solve

two equations.

Holt McDougal Algebra 1

8-8 Completing the Square

Check It Out! Example 3a

Solve by completing the square.

3x2 – 5x – 2 = 0

Step 1 Divide by 3 to make a = 1.

Write in the form x2 + bx = c.

Holt McDougal Algebra 1

8-8 Completing the Square

Complete the square.

Factor and simplify.

Step 3

Step 4

Check It Out! Example 3a Continued

Solve by completing the square.

Step 2 .

Holt McDougal Algebra 1

8-8 Completing the Square

Check It Out! Example 3a Continued

Solve by completing the square.

Write and solve two

equations.

Step 6

−

Take the square root

of both sides.

Step 5

Holt McDougal Algebra 1

8-8 Com