Completing the Square 1.7

What is completing the square? What steps do you follow to complete the

square?

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

Solve:

(a + b)2 =

(a - b)2 =

Make a perfect square trinomial

Find the value of c that makes x2 + 16x + c a perfect square trinomial. Then write the expression as the square of a binomial.SOLUTION

STEP 1Find half the coefficient of x.STEP 2

162 = 8

Square the result of Step 1. 82 = 64STEP 3Replace c with the result of Step 2. x2 + 16x + 64

Then x2 + 16x + 64 = (x + 8)(x + 8) = (x + 8)2

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

x2 + 22x + cx2 11x

11x 121x

x

11

11

SOLUTION

STEP 1

Find half the coefficient of x.STEP 2

222 = 11

Square the result of Step 1. 112 = 121STEP 3

Replace c with the result of Step 2. x2 + 22x + 121The trinomial x2 + 22x + c is a perfect square when c = 121.

Find the value of c that makes x2 – 6x + c

a perfect square trinomial.

Write the expression as the square of a binomial.

2

2

b

c 932

6 22

962 xx

Steps for Completing the Squareax2 + bx + c

1.Make sure the coefficient of the x2 term is one. (If it is not, divide the equation by the coefficient.)

2.Move the constant number to the other side.

3. Divide “b” by 24. Square the result from #2.5. Add this number to both sides of the

equation.6. Factor and solve for x.

Solve ax2 + bx + c = 0 when a = 1Solve 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 + 36to each side.

(x – 6)2 = 32Write 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

Solving a Quadratic equation if the coefficient of x2 is 1

Solve by completing the square.

x2 + 10x -3 = 0

x2 + 10x + ___ = 3 +___

x2 + 10x + 25 = 3 + 25

(x + 5)2 = 282552

10 22

285

285

x

x725x

Solving a Quadratic Equation if the Coefficient of x2 is not 1

Solve by completing the square.

3x2 – 6x + 12 = 0

3x2 − 6x + ___ = −12 + ___3/3x2− 6/3x + ___=−12/3+ ___

x2 −2x + ___=−4 + ___

(−2/2)2=1

x2 −2x + 1=−4 + 1

(x−1)2 = −3

31

3131

ix

xx

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 + ? ) + 22 Prepare to complete the square.

y + 25 = (x2 – 10x + 25) + 22 Add –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

Writing a Quadratic Function in Vertex Form

Write the quadratic function in vertex form. y = x2 -8x +11

y + ___ = (x2 −8x + ___) +11

y + 16 = (x2 −8x + 16) +11

y + 16 = (x−4)2 +11

y = (x−4)2 −5

What is the vertex?

Find the maximum value of a quadratic function

The height y (in feet) of a baseball t seconds after it is hit is given by this function:

Baseball

y = –16t2 + 96t + 3

Find the maximum height of the baseball.

SOLUTION

The maximum height of the baseball is the y-coordinate of the vertex of the parabola with the given equation.

• What is completing the square?Another method for solving a quadratic equations.• What steps do you follow to complete the

square?1.Make sure the coefficient of the x2 term is one. (If

it is not, divide the equation by the coefficient.)2.Move the constant number to the other side.3. Divide “b” by 24. Square the result from #2.5. Add this number to both sides of the equation.6. Factor and solve for x.

Assignment 1.7

p. 54, 3-7 odd,

13-17 odd, 23-27 odd, 39-45 odd, 51