CHAPTER 13: QUADRATIC EQUATIONS AND APPLICATIONS Contents

Chapter 13

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

Contents

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 1.................................. 363 EXERCISE ........................................................................................................................................... 365 SECTION 13.3: QUADRATIC FORMULA ................................................................................................ 366 A. DETERMINANT OF A QUADRATIC EQUATION ......................................................................... 366 B. APPLY THE QUADRATIC FORMULA .......................................................................................... 368 C. MAKE ONE SIDE OF AN EQUATION EQUAL TO ZERO .............................................................. 370 EXERCISE ........................................................................................................................................... 371 SECTION 13.4: APPLICATIONS WITH QUADRATIC EQUATIONS .......................................................... 372 A. PYTHAGOREAN THEOREM ....................................................................................................... 372 B. PROJECTILE MOTION ................................................................................................................ 373 C. COST AND REVENUE................................................................................................................. 374 EXERCISE ........................................................................................................................................... 376 CHAPTER REVIEW ................................................................................................................................. 377

355

Chapter 13

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 is called the leading term, is call the linear term, and 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 . Then = if and only if = ?

In other words, = if and only if = or = -

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) 82 = 648

b) 2 = 75

YOU TRY

Solve. a) 2 = 81

b) 2 = 44

356

MEDIA LESSON Solve equations with even exponents (Duration 4:26)

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

Chapter 13

Consider: 52= ________________ and (-5)2 = ________________________ When we clear an even exponent, we have ________________________________________________.

Example: Solve. a) (5 - 1)2 = 49

b) 4(3 + 2)4 = 81

YOU TRY

Solve. a) ( + 4)2 = 25

b) (6 - 9)2 = 45

357

Chapter 13

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 1st (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) 4 - 2(2 + 1)2 = -46

b) 5(3 - 2)2 + 6 = 46

YOU TRY Solve. a) 5(3x - 6)2 + 7 = 27

b) 5(r + 4)2 + 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 2 = -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

22 + 5 = 4

22 = 4 - 5

22 = -1

2

=

-

1 2

This equation does not have a real solution.

358

Chapter 13

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 term in the middle because if we apply the square root property to the 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 term in the middle and then apply the square root property to solve the equations.

Recall: Perfect square formula + + = ( + ) or - + = ( - )

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) 2 + 8 + 16 = 4

b) 92 - 12 + 4 = 25

YOU TRY Solve. a) 2 - 6 + 9 = 81

b) 92 + 30 + 25 = 4

359

EXERCISE Solve by applying the square root property. 1) ( - 3)2 = 16 3) ( - 7)2 = 4 5) ( + 5)2 = 81 7) ( + 9)2 = 37 9) ( - 9)2 = 63 11) (9 + 1)2 = 9 13) (3 - 6)2 = 25 15) 5( - 5)2 + 13 = 103

17) (2 + 1)2 = 0 19) 32 + 2 = 2 + 24 21) 2( + 9)2 - 19 = 37 23) 7(2 + 6)2 - 5 = 170

2) ( - 2)2 = 49 4) ( - 5)2 = 16 6) ( + 3)2 = 4 8) ( + 5)2 = 57 10) ( + 1)2 = 125 12) (7 - 8)2 = 36 14) 5( - 7)2 - 6 = 369 16) 22 + 7 = 5 18) ( - 4)2 = 25 20) 82 - 29 = 25 + 22 22) 3( - 3)2 + 2 = 164 24) 6(4 - 4)2 - 5 = 145

Chapter 13

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

25) 2 + 12 + 36 = 49

26) 2 + 6 + 9 = 2

27) 162 - 40 + 25 = 16

28) 2 + 4 + 4 = 1

29) 2 - 14 + 49 = 9

30) 252 + 10 + 1 = 49

360

SECTION 13.2: COMPLETING THE SQUARE

Chapter 13

When solving quadratic equations previously (then known as trinomial equations), we factored to solve. However, recall, not all equations are factorable. Consider the equation 2 - 2 - 7 = 0. This equation is not factorable, but there are two solutions to this equation: 1 + 2 and 1 - 2. 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 . + + is easily factored to ________________________________ To make + + a perfect square, = ___________________

Example:

a) 2 + 10 +

b) 2 - 7 +

c)

2

-

3 7

+

d)

2

+

6 5

+

361

 Note

Chapter 13

To complete the square of any trinomial, we always square half of the linear term's coefficient, i.e.,

We usually use the second expression when the middle term's coefficient is a fraction.

YOU TRY

Complete the square by finding :

a) 2 + 8 +

b) 2 - 7 +

c)

2

+

5 3

+

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

Steps to solving quadratic equations by completing the square

Given a quadratic equation + + = , we can use the following method to solve for .

Step 1. Rewrite the quadratic equation so that the coefficient of the leading term is one, and the

original constant coefficient is on the opposite side of the equal sign from the leading and

linear terms.

+ __________ = + _____________

Step 2. If , divide both sides of the equation by

Step

3.

Complete

the

square,

i.e.,

and

add

the

result

to

both

sides

of

the

quadratic

equation.

Step 4. Rewrite the perfect square trinomial in factored form.

Step 5. Solve using the square root property.

Step 6. Verify the solution(s).

MEDIA LESSON Solve quadratic equation by completing the square (Duration 8:40)

View the video lesson, take notes and complete the problems below Solve the quadratic equation using the square root principle.

( - 5)2 = 28

362

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