3.7 Warm Up

Find the vertex and A of S.

1. y = (x – 2)² - 6

2. y = (x + 5)² + 6

3. y = (x – 8)² - 2

4. y = 2(x – 4)(x – 6)

5. y = -(x + 3)(x – 5)

3.7 Complete the Square

2 Reasons to Complete the Square

1. To solve quadratics

2. To write the function from standard to vertex form.

EXAMPLE 1 Solve a quadratic equation by finding square roots

Solve x2 – 8x + 16 = 25.

x2 – 8x + 16 = 25 Write original equation.

(x – 4)2 = 25 Write left side as a binomial squared.

x – 4 = +5 Take square roots of each side.

x = 4 + 5 Solve for x.

The solutions are 4 + 5 = 9 and 4 –5 = – 1.

ANSWER

GUIDED PRACTICE

1. x2 + 6x + 9 = 36.

3 and –9.ANSWER

Solve the equation by finding square roots.

2. x2 – 10x + 25 = 1.

4 and 6.ANSWER

3. x2 – 24x + 144 = 100.

2 and 22.ANSWER

GUIDED PRACTICE for Examples 1 and 2

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

4.

x2 + 14x + c

49 ; (x + 7)2ANSWER

5.

x2 + 22x + c

121 ; (x + 11)2ANSWER

6.

x2 – 9x + c

ANSWER ; (x – )2.814

92

To solve quadratics by completing the square. . .1. Write one side of the equation in the

form x2 + bx (move the c over)

2. Find the term to complete the square and add to both sides

3. When you add (b/2)2, you now can factor it into

4. Then, take the square root to solve.

2

2

b

2

2

bx

EXAMPLE 3 Solve ax2 + bx + c = 0 when a = 1

Solve x2 – 12x + 4 = 0 by completing the square.

x2 – 12x + 4 = 0 Write original equation.

x2 – 12x = –4 Write left side in the form x2 + bx.

x2 – 12x + 36 = –4 + 36 Add –122

2( ) = (–6)2= 36 to each side.

(x – 6)2 = 32 Write left side as a binomial squared.

Solve for x.

Take square roots of each side.x – 6 = + 32

x = 6 + 32

x = 6 + 4 2 Simplify: 32 = 16 2 = 4 2

The solutions are 6 + 4 and 6 – 42 2ANSWER

EXAMPLE 4 Solve ax2 + bx + c = 0 when a = 1

Solve 2x2 + 8x + 14 = 0 by completing the square.

2x2 + 8x + 14 = 0 Write original equation.

x2 + 4x + 7 = 0

Write left side in the form x2 + bx.

x2 + 4x + 4 = –7 + 4 Add 42

2( ) = 22 = 4 to each side.

(x + 2)2 = –3 Write left side as a binomial squared.

Solve for x.

Take square roots of each side.x + 2 = + –3

x = –2 + –3

x = –2 + i 3

x2 + 4x = –7

Divide each side by the coefficient of x2.

Write in terms of the imaginary unit i.

The solutions are –2 + i 3 and –2 – i 3 .

GUIDED PRACTICE

x2 + 6x + 4 = 0

–3+ 5ANSWER

7.

Solve the equation by completing the square.

x2 – 10x + 8 = 0

5 + 17ANSWER

8.

2n2 – 4n – 14 = 0

1 + 2 2ANSWER

9.

3x2 + 12x – 18 = 010.

–2 + 10ANSWER

11. 6x(x + 8) = 12

–4 +3 2ANSWER

1 + 26ANSWER

12. 4p(p – 2) = 100

EXAMPLE 6 Write a quadratic function in vertex form

Write y = x2 – 10x + 22 in vertex form. Then identify the vertex.

y = x2 – 10x + 22 Write original function.

y + ? = (x2 –10x + ? ) + 22Prepare to complete the square.

y + 25 = (x2 – 10x + 25) + 22Add –102

2( ) = (–5)2= 25 to each side.

y + 25 = (x – 5)2 + 22 Write x2 – 10x + 25 as a binomial squared.

y = (x – 5)2 – 3 Solve for y.

The vertex form of the function is y = (x – 5)2 – 3. The vertex is (5, –3).

ANSWER

GUIDED PRACTICE

y = x2 – 8x + 17

y = (x – 4)2 + 1 ; (4, 1).ANSWER

13.

Write the quadratic function in vertex form. Then identify the vertex.

y = x2 + 6x + 3

y = (x + 3)2 – 6 ; (–3, –6)

ANSWER

14.

f(x) = x2 – 4x – 4

y = (x – 2)2 – 8 ; (2 , –8)ANSWER

15.