Copyright © Cengage Learning. All rights reserved.

Quadratic Equations, Quadratic Functions, and Complex Numbers 9

Copyright © Cengage Learning. All rights reserved.

Section 9.29.2

Solving Quadratic Equations by Completing the Square

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Objectives

1. Complete the square of a binomial to create a perfect trinomial square.

2. Solve a quadratic equation by completing the square.

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Solving Quadratic Equations by Completing the Square

When the polynomial in a quadratic equation does not factor easily, we can solve the equation by using a method called completing the square. In fact, we can solve any quadratic equation by completing the square.

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Complete the square of a binomial to create a perfect trinomial square

1.

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Complete the square of a binomial to create a perfect trinomial square

The method of completing the square is based on the following special products:

x2 + 2bx + b2 = (x + b)2 and x2 – 2bx + b2 = (x – b)2

Recall that the trinomials x2 + 2bx + b2 and x2 – 2bx + b2 are both perfect trinomial squares, because each one factors as the square of a binomial.

In each trinomial, if we take one-half of the coefficient of x and square the result, we get the third term.

and

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To form a perfect trinomial square from the binomialx2 + 12x, we take one-half of the coefficient x of (the 12), square the result, and add it to x2 + 12x.

This result is a perfect trinomial square, because

x2 + 12x + 36 = (x + 6)2.

Complete the square of a binomial to create a perfect trinomial square

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Example

Form perfect trinomial squares using

a. x2 + 4x b. x2 – 6x c. x2 – 5x

Solution:

a.

b.

This is (x + 2)2.

This is (x – 3)2.

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Example – Solution

c.

In each case, note that of the coefficient of x is the

second term of the binomial factorization.

cont’d

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Solve a quadratic equation by completing the square

2.

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Solve a quadratic equation by completing the square

If the quadratic equation ax2 + bx + c = 0 has a leading coefficient of 1 (a = 1) and especially if the middle term iseven, we can solve by completing the square fairly easily.

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Exampl

Solve by completing the square: x2 – 8x – 20 = 0

Solution:

We can solve the equation by completing the square.

x2 – 8x – 20 = 0

x2 – 8x = 20

We then find one-half of the coefficient of x, square the result, and add it to both sides to make the left side a trinomial square.

Add 20 to both sides.

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Example – Solution

x2 – 8x + 16 = 20 + 16

(x – 4)2 = 36

x – 4 =

x = 4 6

Simplify.

Factor x2 – 8x + 16 and simplify.

Use the square root property to solve for x – 4.

Add 4 to both sides,

cont’d

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Example – Solution

Because of the sign, there are two solutions.

x = 4 + 6 or x = 4 – 6

x = 10 = –2

Check each solution. Note that this example could be solved by factoring.

cont’d

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To solve a quadratic equation by completing the square, we follow these steps.

Completing the Square

1. If necessary, write the quadratic equation in quadratic form, ax2 + bx + c = 0.

2. If the coefficient of x2 is not 1, divide both sides of the equation by a, the coefficient of x2.

3. If necessary, add a number to both sides of the equation to place the constant term on the right side of the equal sign.

Solve a quadratic equation by completing the square

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4. Complete the square:

a. Find one-half of the coefficient of x and square it.

b. Add the square to both sides of the equation.

5. Factor the perfect trinomial square on the left side of the equation and combine any like terms on the right side of the equation.

6. Use the square-root property to solve the resulting quadratic equation.

7. Check each solution in the original equation.

Solve a quadratic equation by completing the square

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Example

Solve by completing the square: 4x2 –3 = –4x

Solution:

We first write the equation in quadratic form

4x2 + 4x – 3 = 0

and then divide every term on both sides of the equation by 4 so that the coefficient of x2 is 1.

Add 4x to both sides.

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Example – Solution

We then use completing the square to solve the equation.

Simplify.

Divide both sides by 4.

Add to both sides.

Add to both sides to

complete the square.

cont’d

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Example – Solution

Check each solution. Note that this equation also can be solved by factoring.

cont’d

Factor and simplify.

Solve for .

Add to both sides.

or