introduction a trinomial of the form that can be written as the square of a binomial is called a...

Introduction

A trinomial of the form that can be written

as the square of a binomial is called a perfect square

trinomial. We can solve quadratic equations by

transforming the left side of the equation into a perfect

square trinomial and using square roots to solve.

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5.2.3: Completing the Square

Key Concepts• When the binomial (x + a) is squared, the resulting

perfect square trinomial is x2 + 2ax + a2. • When the binomial (ax + b) is squared, the resulting

perfect square trinomial is a2x2 + 2abx + b2.

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Key Concepts, continued

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Completing the Square to Solve Quadratics

1. Make sure the equation is in standard form, ax2 + bx + c.

2. Subtract c from both sides. 3. Divide each term by a to get a leading

coefficient of 1. 4. Add the square of half of the coefficient of the

x-term to both sides to complete the square. 5. Express the perfect square trinomial as the

square of a binomial. 6. Solve by using square roots.

Common Errors/Misconceptions• neglecting to add the c term of the perfect square

trinomial to both sides

• not isolating x after squaring both sides

• forgetting that, when taking the square root, both the positive and negative roots must be considered (±)

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Guided Practice

Example 2Solve x2 + 6x + 4 = 0 by completing the square.

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Guided Practice: Example 2, continued

1. Determine if x2 + 6x + 4 is a perfect square trinomial. Take half of the value of b and then square the result. If this is equal to the value of c, then the expression is a perfect square trinomial.

x2 + 6x + 4 is not a perfect square trinomial because the square of half of 6 is not 4.

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2. Complete the square.

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x2 + 6x + 4 = 0 Original equation

x2 + 6x = –4 Subtract 4 from both sides.

x2 + 6x + 32 = –4 + 32

Add the square of half of the coefficient of the x-term to both sides to complete the square.

x2 + 6x + 9 = 5 Simplify.


3. Express the perfect square trinomial as the square of a binomial. Half of b is 3, so the left side of the equation can be written as (x + 3)2.

(x + 3)2 = 5

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4. Isolate x.

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(x + 3)2 = 5 Equation

Take the square root of both sides.

Subtract 3 from both sides.


5. Determine the solution(s). The equation x2 + 6x + 4 = 0 has two solutions,

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Guided Practice

Example 3Solve 5x2 – 50x – 120 = 0 by completing the square.

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1. Determine if 5x2 – 50x – 120 = 0 is a perfect square trinomial. The leading coefficient is not 1.

First divide both sides of the equation by 5 so that a = 1.

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5x2 – 50x – 120 = 0 Original equation

x2 – 10x – 24 = 0 Divide both sides by 5.

Guided Practice: Example 3, continuedNow that the leading coefficient is 1, take half of the value of b and then square the result. If the expression is equal to the value of c, then it is a perfect square trinomial.

5x2 – 50x – 120 = 0 is not a perfect square trinomial because the square of half of –10 is not –24.

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2. Complete the square.

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x2 – 10x – 24 = 0 Equation

x2 – 10x = 24 Add 24 to both sides.

x2 – 10x + (–5)2 = 24 + (–5)2

Add the square of half of the coefficient of the x-term to both sides to complete the square.

x2 – 10x + 25 = 49 Simplify.


3. Express the perfect square trinomial as the square of a binomial. Half of b is –5, so the left side of the equation can be written as (x – 5)2.

(x – 5)2 = 49

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4. Isolate x.

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(x – 5)2 = 49 Equation

Take the square root of both sides.

Add 5 to both sides.

x = 5 + 7 = 12 orx = 5 – 7 = –2

Split the answer into two separate equations and solve for x.


5. Determine the solution(s). The equation 5x2 – 50x – 120 = 0 has two solutions, x = –2 or x = 12.

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introduction a trinomial of the form that can be written as the square of a binomial is called a...

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