[PDF] CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS PDF Chapter%2013%20-Quadratic%20equations.pdf

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[PDF] CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS

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Chapter 13

355

CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS

Chapter Objectives

By the end of this chapter, students should be able to: Apply the Square Root Property to solve quadratic equations Solve quadratic equations by completing the square and using the Quadratic Formula Solve applications by applying the quadratic formula or completing the square

CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS .................................................................. 355

SECTION 13.1: THE SQUARE ROOT PROPERTY .................................................................................... 356

A. SOLVE BASIC QUADRATIC EQUATIONS USING SQUAREROOT PROPERTY ............................. 356

B. ISOLATE THE SQUARED TERM .................................................................................................. 358

C. USE THE PERFECT SQUARE FORMULA ..................................................................................... 359

EXERCISE ........................................................................................................................................... 360

SECTION 13.2: COMPLETING THE SQUARE .......................................................................................... 361

A. COMPLETE THE SQUARE .......................................................................................................... 361

B. SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE, a = 1 .................................. 362

C. SOLVE QUADRATIC EQUATIONS BY COMPLETING THE SQUARE, A т 1 .................................. 363

EXERCISE ........................................................................................................................................... 365

SECTION 13.3: QUADRATIC FORMULA ................................................................................................ 366

A. DETERMINANT OF A QUADRATIC EQUATION ......................................................................... 366

B. APPLY THE QUADRATIC FORMULA .......................................................................................... 368

MAKE ONE SIDE OF AN EQUATION EQUAL TO ZERO .............................................................. 370

EXERCISE ........................................................................................................................................... 371

SECTION 13.4: APPLICATIONS WITH QUADRATIC EQUATIONS .......................................................... 372

A. PYTHAGOREAN THEOREM ....................................................................................................... 372

PROJECTILE MOTION ................................................................................................................ 373

C. COST AND REVENUE ................................................................................................................. 374

EXERCISE ........................................................................................................................................... 376

CHAPTER REVIEW ................................................................................................................................. 377

Chapter 13

356

We might recognize a quadratic equation from the factoring chapter as a trinomial equation. Although,

it may seem that they are the same, but they aren't the same. Trinomial equations are equations with any

three terms. These terms can be any three terms where the degree of each can vary. On the other hand,

quadratic equations are equations with specific degree on each term.

Definition

A quadratic equation is a polynomial equation of the form

Where ࢇ࢞

is called the leading term, ࢈࢞ is call the linear term, and ࢉ is called the constant coefficient

(or constant term). Additionally,

SECTION 13.1: THE SQUARE ROOT PROPERTY

A. SOLVE BASIC QUADRATIC EQUATIONS USING SQUAREROOT PROPERTY

Square root property

Let ࢞൒૙ and ࢇ൒૙. Then = if and only if ࢞=±ξࢇ

In other words,

=ࢇ if and only if ࢞=ξࢇ or ࢞= െξࢇ

MEDIA LESSON

Solve basic quadratic equations using square root property (Duration 2:53) View the video lesson, take notes and complete the problems below

Example:

a) 8ݔ =648 b) ݔ =75

YOU TRY

Solve.

a) ݔ =81 b) ݔ =44

Chapter 13

357

MEDIA LESSON

Solve equations with even exponents (Duration 4:26) View the video lesson, take notes and complete the problems below

Consider: 5

= ________________ and (െ5) = ________________________ When we clear an even exponent, we have ________________________________________________.

Example: Solve.

a) (5ݔെ1) = 49 =81

YOU TRY

Solve.

a) (ݔ+4) =25 b) (6ݔെ9) =45

Chapter 13

358

B. ISOLATE THE SQUARED TERM

Let's look at examples where the leading term, or squared term, is not isolated. Recall, the squared term

must be isolated to apply the square root property.

MEDIA LESSON

Solve equations using square root property - Isolating the squared term 1 st (Duration 5:00) View the video lesson, take notes and complete the problems below Before we can clear an exponent, it must first be _____________________________.

Example:

a) 4െ2(2ݔ+1) =െ46 b) 5(3ݔെ2) +6=46

YOU TRY

Solve.

a) 5(3xെ6) +7=27 b) 5(r+4) +1=626 Note: When we have the other side of the equation of a squared term negative, the equation does not have a real solution. For example, the equation ݔ =െ1 does not have a real solution. There is a complex solution for this equation but we will not discuss it in this class.

Example:

Solve 2݊ +5=4 2݊ =4െ5 2 =െ1 =െ1 2

This equation does

not have a real solution.

Chapter 13

359

C. USE THE PERFECT SQUARE FORMULA

In order for us to be able to apply the square root property to solve a quadratic equation, we cannot have

the ݔ term in the middle because if we apply the square root property to the ݔ term, we will make the

equation more complicated to solve. However sometimes, we have special cases that we can apply the perfect square formula to get rid of the ݔ term in the middle and then apply the square root property to solve the equations.

Recall: Perfect square formula

or ࢇ

MEDIA LESSON

Solve equations using square root property - Perfect Square formula (Duration 4:09) View the video lesson, take notes and complete the problems below

Example:

a) ݔ +8ݔ+16=4 b) 9ݔ െ12ݔ+4=25

YOU TRY

Solve.

a) ݔ െ6ݔ+9=81 b) 9ݔ +30ݔ+25=4

Chapter 13

360

EXERCISE

Solve by applying the square root property.

1) (ݔെ3)

=16

2) (ݔെ2)

=49

3) (ݔെ7)

=4 =16

5) (݌+5)

=81 =4

7) (ݐ+9)

=37

8) (ܽ

=57

9) (݊െ9)

=63

10) (ݎ+1)

=125

11) (9ݎ+1)

12) (7݉െ8)

=36 =25

14) 5(݇െ7)

െ6=369

15) 5(݃െ5)

+13=103

16) 2݊

+7=5 =0

18) (ݖെ4)

=25

19) 3݊

+2݊=2݊+24

20) 8݊

െ29=25+2݊

21) 2(ݎ+9)

െ19=37

22) 3(݊െ3)

+2=164

23) 7(2ݔ+6)

െ5=170

24) 6(4ݔെ4)

െ5=145 Apply the perfect square formula and solve the equations by using the square root property.

25) ݔ

+12ݔ+36=49

26) ݔ

+6ݔ+9=2

27) 16ݔ

െ40ݔ+25=16

28) ݔ

+4ݔ+4=1

29) ݔ

െ14ݔ+49=9

30) 25ݔ

+10ݔ+1=49

Chapter 13

361

SECTION 13.2: COMPLETING THE SQUARE

When solving quadratic equations previously (then known as trinomial equations), we factored to solve.

However, recall, not all equations are factorable. Consider the equation ݔ െ2ݔെ7=0. This equation is not factorable, but there are two solutions to this equation: 1+ξ2 and 1െξ2. Looking at the form of these solutions, we obtained these types of solutions in the previous section while using the square

root property. If we can obtain a perfect square, then we can apply the square root property and solve as

usual. This method we use to obtain a perfect square is called completing the square. Recall. Special product formulas for perfect square trinomials: We use these formulas to help us solve by completing the square.

A. COMPLETE THE SQUARE

We first begin with completing the square and rewriting the trinomial in factored form using the perfect

square trinomial formulas.

MEDIA LESSON

Complete the square (Duration 5:00)

View the video lesson, take notes and complete the problems below

Complete the square. Find ࢉ.

is easily factored to ________________________________

To make

+࢈࢞+ࢉ a perfect square, ࢉ= ___________________

Example:

a) ݔ +10ݔ+ܿ b) ݔ െ7ݔ+ܿ c)

Chapter 13

362
Note

To complete the square of any trinomial, we

always square half of the linear term's coefficient, i.e., p quotesdbs_dbs20.pdfusesText_26

[PDF] [PDF] CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS

Chapter 13

CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS

Chapter Objectives

Contents

Chapter 13

Definition

Where ࢇ࢞

SECTION 13.1: THE SQUARE ROOT PROPERTY

Square root property

In other words,

MEDIA LESSON

Example:

YOU TRY

Solve.

Chapter 13

MEDIA LESSON

Consider: 5

Example: Solve.

YOU TRY

Solve.

Chapter 13

B. ISOLATE THE SQUARED TERM

MEDIA LESSON

Example:

YOU TRY

Solve.

Example:

This equation does

Chapter 13

C. USE THE PERFECT SQUARE FORMULA

Recall: Perfect square formula

MEDIA LESSON

Example:

YOU TRY

Solve.

Chapter 13

EXERCISE

Solve by applying the square root property.

1) (ݔെ3)

2) (ݔെ2)

3) (ݔെ7)

5) (݌+5)

7) (ݐ+9)

8) (ܽ

9) (݊െ9)

10) (ݎ+1)

11) (9ݎ+1)

12) (7݉െ8)

14) 5(݇െ7)

15) 5(݃െ5)

16) 2݊

18) (ݖെ4)

19) 3݊

20) 8݊

21) 2(ݎ+9)

22) 3(݊െ3)

23) 7(2ݔ+6)

24) 6(4ݔെ4)

25) ݔ

26) ݔ

27) 16ݔ

28) ݔ

29) ݔ

30) 25ݔ

Chapter 13

SECTION 13.2: COMPLETING THE SQUARE

A. COMPLETE THE SQUARE

MEDIA LESSON

Complete the square (Duration 5:00)

Complete the square. Find ࢉ.

To make

Example:

Chapter 13

To complete the square of any trinomial, we