Lesson 9.6 Solve Quadratic Equations by the Quadratic FormulaEssential Question: How do you solve quadratic equations by using the quadratic formula? Day 3

CC.9.12.A.REI.4bSolve quadratic equations by inspection, by taking square roots, completing thesquare, the quadraticformula and factoring, as appropriate to theinitial form of theequation. Recognizewhen the quadratic formula gives complexsolutions.

Warm-up:

Date: 4-12-13

Get out Homework 9-6 Practice BAnswers1. -13.10, 6.102. -2.15, 2.483. -1.82, 2.074. -3.73, -0.275. -3, 46. -2.61, 1.287. -4.61, -1.39

8. -13.44, 7.449. -2.11, 2.3610. -1.45, 1.2511. -3/2, 1/312. no solution

EXAMPLE 1 Standardized Test Practice

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

–1 and – A 83

B –1 and 83 C 1 and – 8

3 D 1 and 83

SOLUTION

Write original equation.3x2 + 5x = 8

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

Quadratic formula ±x = –b b2 – 4ac2a

EXAMPLE 1 Standardized Test Practice

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

= – 83

The correct answer is C. ANSWER

x = –5 ± 52 – 4(3)(–8)

2(3)

EXAMPLE 2 Solve a quadratic equation

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

EXAMPLE 2 Solve a quadratic equation

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.

ANSWER

1 + 57 2.14 and 1 – 4

–1.64.4

The solutions are

CHECK

57

GUIDED PRACTICE for Examples 1and 2

Solve the equation by Quadratic Formula.

1. x2 – 8x + 16 = 0

ANSWER 4

2. 3n2 – 5n = –1

ANSWER 0.23, 1.43

3. 4z2 = 7z + 2

ANSWER 0.25, 2

EXAMPLE 3 Use the quadratic formula

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

EXAMPLE 3 Use the quadratic formula

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

20,83694 + –20

= Simplify.

The solutions of the equation are:

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

x = (–94)2 – 4 (10)(–300)–(–94)

+ –2(10)

94 + 20,836

2012 94 – 20,836

20–3.and

GUIDED PRACTICE for Example 3

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

4.

ANSWER 1986

To find:

EXAMPLE 4 Choose a solution method

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.

Choose a solution methodEXAMPLE 4

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.

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

b. x2 + 4x = 0

SOLUTION

Choose a solution methodEXAMPLE 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.

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

c. 5x2 + 9x – 4 = 0

SOLUTION

GUIDED PRACTICE for Example 4

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

SOLUTION

Factoring because the expression factors easily.

5. x2 + x – 6 = 0

GUIDED PRACTICE for Example 4

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

SOLUTION

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

6. x2 – 9 = 0

GUIDED PRACTICE for Example 4

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

SOLUTION

Completing 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.

7. x2 + 6x = 5

Classwork/Homework

• 9.6 Practice B Finish all!