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In Chapter 8we revisit quadratic equations. In earlier chapters, we solved quadratic equations by factoring and applying the zero product rule. In this chapter, we present two additional techniques to solve quadratic equations. However, these techniques can be used to solve quadratic equations even if the equations are not factorable. Complete the word scramble to familiarize yourself with the key terms for this chapter. As a hint, there is a clue for each word.

Quadratic

Equations

and Functions

8.1Square Root Property and Completing the Square

8.2Quadratic Formula

8.3Equations in Quadratic Form

8.4Graphs of Quadratic Functions

8.5Vertex of a Parabola and Applications551

88

1.raeqsu( __________ root property)

2.eitgcompln( __________ the square)

3.dqatriuac( ________ formula)

4.taiicidsrmnn

5.aobrplaa(shape of a quadratic function)

6.xteerv(maximum or minimum point on the graph of a parabola)

7.sxia( ________ of symmetry)

1b 2 ?4ac2 IAmiL2872X_ch08_551-618 9/30/06 06:46 AM Page 551CONFIRMING PAGES

1. Solving Quadratic Equations by Using

the Square Root Property In Section 5.8 we learned how to solve a quadratic equation by factoring and ap- plying the zero product rule. For example,

Set one side equal to zero.

Factor.

orSet each factor equal to zero. or

The solutions are and

It is important to note that the zero product rule can only be used if t he equa- tion is factorable. In this section and Section 8.2, we will learn how to solve quad- ratic equations, factorable and nonfactorable.

The first technique utilizes the

square root property.x??9.x?9x??9x?9x?9?0x?9?0 1x?921x?92?0x 2 ?81?0x 2 ?81

The Square Root Property

For any real number,k,if .

Note:The solution may also be written as read xequals "plus or minus the square root of k x??1k, x 2 ?k, then x?1k or x??1k

Section 8.1Square Root Property and Completing

the Square

Concepts

1.Solving Quadratic Equations by

Using the Square Root Property

2.Solving Quadratic Equations by

Completing the Square

3.Literal Equations

Avoiding Mistakes:

A common mistake is to forget

the symbol when solving the equation : x??1k x 2 ?k?

Solving Quadratic Equations by Using

the Square Root Property Use the square root property to solve the equations. a.b.c.

Solution:

a.The equation is in the form

Apply the square root property.

The solutions are Notice that this is the same solution obtained by factoring and applying the zero product rule. b.Rewrite the equation to fit the form

The equation is now in the form

Apply the square root property.

??5ix??1?25 x 2 ?k.x 2 ??253x 2 ??75x 2 ?k.3x 2 ?75?0x?9 and x??9. x??9 x??181 x 2 ?k.x 2 ?811w?32 2 ?203x 2 ?75?0x 2 ?81

Example 1

552Chapter 8Quadratic Equations and Functions

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The solutions are .

Check :Check: c.The equation is in the form where

Apply the square root property.

Simplify the radical.

Solve for

w

The solutions are

Use the square root property to solve the equations.

1.2.3.

2. Solving Quadratic Equations by Completing

the Square In Example 1(c) we used the square root property to solve an equation where the square of a binomial was equal to a constant. The square of a binomial is the factored form of a perfect square trinomi al. For example:

Perfect Square

TrinomialFactored Form

For a perfect square trinomial with a leading coefficient of 1, the constant term is the square of one-half the linear term coefficient. For example: In general an expression of the form is a perfect square trinomial if .The process to create a perfect square trinomial is called completing the square .n?1 1 2 b2 2 x 2 ?bx?n3 1 2 11024
2 x 2 ?10x?251p?72 2 p 2 ?14p?491t?32 2 t 2 ?6t?91x?52 2 x 2 ?10x?251w?32 2 ?20 1t?52 2 ?188x 2 ?72?0a 2 ?100

Skill Practice

w??3?215 and w??3?215.w??3 ? 215 w?3??215 w?3??22 2 ?5 w?3??120 x?1w?32.x 2 ?k,1w?32 2 ?20?75?75?0?75?75?0

31?252?75?031?252?75?03125i

2

2?75?03125i

2

2?75?031

5 i2 2 ?75?0315i2 2 ?75?03x 2 ?75?03x 2 ?75?0x??5ix?5ix?5i and x??5i

Skill Practice Answers

1.2.

3.t?5?312x??3ia??10

square of a binomialconstant Section 8.1Square Root Property and Completing the Square553 IA miL2872X_ch08_551-618 9/30/06 06:46 AM Page 553

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Completing the Square

Determine the value of nthat makes the polynomial a perfect square trinomial. Then factor the expression as the square of a binomial. a.b. c.d.

Solution:

The expressions are in the form The value of nequals the square of one-half the linear term coefficient a.

Factored form

b.

Factored form

c.

Factored form

d.

Factored form

Determine the value of

nthat makes the polynomial a perfect square trinomial.Then factor. 4.5. 6.7. The process of completing the square can be used to write a quadratic equ a- tion in the form Then the square root prop- erty can be used to solve the equation. The following steps outline the procedure.1x?h2 2 ?k.1a?02ax 2 ?bx?c?0 w 2 ?7 3w? na 2 ?5a?n y 2 ?16y?nx 2 ?20x?n

Skill Practice

ax?2 7b 2 n?3 1 2 1? 4 7 24
2 ?1? 2 7 2 2 4 49
x 2 ?4 7 x? 4 49
x 2 ?4 7 x?n ax?11 2b 2 n?3 1 2 11124
2 ?1 11 2 2 2 121
4 x 2 ?11x? 121
4 x 2 ?11x?n

1x?132

2 n?3 1 2

1?2624

2 ?1?132 2 ?169.x 2 ?26x?169 x 2 ?26x?n 1x?62 2 n?3 1 2 11224
2 ?162 2 ?36x 2 ?12x?36 x 2 ?12x?n 1 1 2 b2 2 .x 2 ?bx?n.x 2 ?4 7x? nx 2 ?11x?n x 2 ?26x?nx 2 ?12x?n

Example 2

Skill Practice Answers

4. 5. 6.

7.n?49

36; aw?76b

2 n?25

4; aa?52b

2 n?64; 1y?82 2 n?100; 1x?102 2

554Chapter 8Quadratic Equations and Functions

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Solving a Quadratic Equation in the Form ax

2 ?bx?c?0(a?0) by Completing the Square and Applying the Square Root Property

1.Divide both sides by ato make the leading coefficient 1.

2.Isolate the variable terms on one side of the equation.

3.Complete the square. (Add the square of one-half the linear term

coefficient to both sides of the equation. Then factor the resulting perfect square trinomial.)

4.Apply the square root property and solve for x.

Solving Quadratic Equations by Completing the

Square and Applying the Square Root Property

Solve the quadratic equations by completing the square and applying the square root property. a.b.

Solution:

a.

Step 1:Since the leading coefficient

ais equal to 1, we do not have to divide by a. We can proceed to step 2.

Step 2:Isolate the variable terms onone side.

Step 3:To complete the square, add

to both sides of the equation.

Factor the perfect square

trinomial.

Step 4:Apply the square root property.

Simplify the radical.

Solve for

x The solutions are imaginary numbers and can be written as and b. x 2 ?13 2 x?15 2?0 4 x 2

4?26x4?304?04 4x

2 ?26x?30?0 4xquotesdbs_dbs21.pdfusesText_27

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