10.6

Warm UpWarm Up

Lesson QuizLesson QuizLesson PresentationLesson Presentation

Solve Quadratic Equations by the Quadratic Formula

10.6 Warm-Up

Evaluate the expression for the given value of x.

1. 15 – (–x) + 9; x = –2

2. 14 – x + 3; x = 8

ANSWER 22

ANSWER 9

ANSWER 22 ft

A basketball is thrown in the air from a height of 6 feet with an initial velocity of 40 feet per second. What is the height (in feet) of the ball after 2 seconds?

3.

10.6 Example 1

What are the solutions of 3x2 + 5x = 8?

–1 and – A 83 B –1 and 8

3 C 1 and – 83 D 1 and 8

3

SOLUTION

Write original equation.3x2 + 5x = 8

Write in standard form.3x2 + 5x – 8 = 0

Quadratic formula ±x = –b b2 – 4ac2a

10.6 Example 1

Substitute values in the quadratic formula: a = 3, b = 5, and c = –8.

Simplify.= –5± 1216

Simplify the square root.= –5 ± 116

The solutions of the equation are –5 + 116 = 1 and

–5 – 116 = – 8

3

x = –5 ± 52 – 4(3)(–8)2(3)

The correct answer is C. ANSWER

10.6 Example 2

2x2 – 7 = x

2x2 – x – 7 = 0

Solve 2x2 – 7 = x.

Write original equation.

Write in standard form.

Quadratic formulax =

b2 – 4ac+ ––b2a

Substitute values in the quadratic formula: a = 2, b = –1, and c = –7.

Simplify.

– (–1) –+ ( –1)2 – 4(2)(–7)

2(2)=

4=+ –1 57

10.6 Example 2

ANSWER

CHECK

1 + 57 2.14 and 1 – 4

–1.64.4

The solutions are 57

Write the equation in standard form, 2x2 – x – 7 = 0. Then graph the related function y = 2x2 – x – 7. The x-intercepts are about –1.6 and 2.1. So, each solution checks.

10.6 Guided Practice

Use the quadratic formula to solve the equation. Round your solution to the nearest hundredth, if necessary.

1. x2 – 8x + 16 = 0

ANSWER 4

2. 3n2 – 5n = –1

ANSWER 0.23, 1.43

3. 4z2 = 7z + 2

ANSWER 0.25, 2

10.6 Example 3

y = 10x2 – 94x + 3900

4200 = 10x2 – 94x – 3900

0 = 10x2 – 94x – 300

Write function.

Substitute 4200 for y.

Write in standard form.

SOLUTION

For the period 1971 2001, the number y of films produced in the world can be modeled by the function y = 10x2 94x + 3900 where x is the number of years since 1971. In what year were 4200 films produced?

FILM PRODUCTION

10.6 Example 3

Substitute values in the quadratic formula: a = 10, b = –94, and c = –300.

20,83694 + –20

= Simplify.

ANSWER There were 4200 films produced about 12 years after 1971, or in 1983.

x = (–94)2 – 4 (10)(–300)–(–94)

+ –2(10)

The solutions of the equation are:

94 + 20,83620

12 94 – 20,83620

–3.and

10.6 Guided Practice

WHAT IF? In Example 3, find the year when 4750 films were produced.

4.

ANSWER 1986

10.6 Example 4

Tell what method you would use to solve the quadraticequation. Explain your choice(s).

a. 10x2 – 7 = 0

SOLUTION

a. The quadratic equation can be solved using square roots because the equation can be written in the form x2 = d.

b. x2 + 4x = 0

b. The equation can be solved by factoring because the expression x2 + 4x can be factored easily. Also, the equation can be solved by completing the square because the equation is of the form ax2 + bx + c = 0 where a = 1 and b is an even number.

c. 5x2 + 9x – 4 = 0

10.6 Example 4

c. The quadratic equation cannot be factored easily, and completing the square will result in many fractions. So, the equation can be solved using the quadratic formula.

SOLUTION

10.6 Guided Practice

Tell what method you would use to solve the quadraticequation. Explain your choice(s).5. x2 + x – 6 = 0

ANSWER Factoring because the expression factors easily.

6. x2 – 9 = 0

ANSWERFactoring because the expression factors easily. Using square roots is another option since the equation can be written in the form x2 = d.

10.6 Guided Practice

7. x2 + 6x = 5

ANSWERCompleting the square because the equation is of the form ax2 + bx = c where a = 1 and b is an even number. Another method is the quadratic formula since the equation does not factor easily.

Tell what method you would use to solve the quadraticequation. Explain your choice(s).

10.6 Lesson Quiz

1. 6x2 – 6 = –5x

Use the quadratic formula to solve the equation. Round your solutions to the nearest hundredth, if necessary.

2. 2x2 + 3x – 8 = 0

–2.87, 1.39ANSWER

,ANSWER 32– 2

3

3. 3m2 – m + 9 = 12

ANSWER –0.85, 1.18

10.6 Lesson Quiz

You have seen that the function y = 10x2 94x + 3900 models the number y of films produced in the world, where x is the number of years since 1971. In what year were 10,000 films produced?

4.

ANSWER about 29.8 years after 1971, or during the year 2000