340 UNIT 5 • Nonlinear Functions and Equations

Summarize the Mathematics

In this investigation, you discovered strategies for expanding and factoring expressions that represent quadratic functions.a How do you go about expanding a product of two linear expressions like (x + a)(x - b)?

b What shortcut can be applied to expand products in the form (x + a)2? In the form (x + a)(x - a)?

c How would your answers to Parts a and b change if the coefficients of x were numbers other than 1?

d How do you go about finding a factored form for quadratic expressions like x2 + bx + c?

e How would you modify your strategy in Part d for quadratic expressions of the form ax2 + bx + c?

Be prepared to explain your ideas and strategies to others in your class.

Check Your UnderstandingCheck Your UnderstandingWrite the following expressions in equivalent expanded or factored forms.

a. (x + 7)(x - 4) b. (x - 5)(x + 5)

c. (x - 3)2 d. x2 - 100

e. x2 + 9x + 20 f. x2 + 3x - 10

g. x2 + 7x h. 5x(x - 3)

IInvest invest iggationation 44 Solving Quadratic Equations Solving Quadratic EquationsIn situations that involve quadratic functions, the interesting questions often require solving equations. For example,

When a pumpkin is dropped from a point 50 feet above the ground, it will hit the ground at the time t that satisfies the equation 50 - 16t2 = 0.

To find points where the main cable of a suspension bridge is 20 feet above the bridge surface, you might need to solve an equation like

0.02x2 - x + 110 = 20.

You can always estimate solutions for these equations by scanning tables of (x, y) values or by tracing coordinates of points on function graphs. In some cases, you can get exact solutions by reasoning algebraically—without use of calculator tables or graphs. As you work on the problems of this investigation, look for answers to this question:

What strategies can be used to solve quadratic equations by factoring and the quadratic formula?

Name:___________________________Unit 2

Lesson 1Investigation 4

LESSON 1 • Quadratic Functions, Expressions, and Equations 341

1 Find, if possible, exact solutions for each of the following equations algebraically. Record steps in your reasoning so that someone else could retrace your thinking.

a. 5x2 + 12 = 57 b. -5x2 + 12 = -33

c. 3x2 - 15 = 70 d. 8 - 2x2 = x2 + 5

e. 7x2 - 24 = 18 - 2x2 f. x2 + 4 = 2x2 + 9

What general guidelines would you suggest for solving equations like these?

2 Solve the following equations algebraically. Record steps in your reasoning so that someone else could retrace your thinking.

a. x(5 - x) = 0 b. 5x2 + 15x = 0

c. 16x - 4x2 = 0 d. 3x + 5x2 = -7x

What general guidelines would you suggest for solving equations like these?

3 Each of the following equations involves a quadratic expression that can be factored into a product of two linear expressions. Solve each equation and record steps in your reasoning so that someone else could retrace your thinking.

a. x2 + 2x - 24 = 0 b. x2 - 6x + 5 = 0

c. -x2 + 8x - 15 = 0 d. x2 + 10x + 21 = 0

e. 2x2 - 12x + 18 = 0 f. x2 + 6x - 16 = 0

When you know several strategies for solving different kinds of equations, the key step in working on any particular problem is matching your strategy to the equation form. Use what you have learned from work on Problems 1–3 to apply effective strategies for solving the quadratic equations in Problem 4.

4 Solve each of these equations by algebraic reasoning. Record steps in your reasoning so that someone else could retrace your thinking. Be prepared to explain how you analyzed each given problem to decide on a solution strategy.

a. x2 + 6x + 5 = 0 b. 6x + x2 = 0

c. x2 + 12x + 20 = 0 d. 7x + x2 + 12 = 0

e. 9 = -7 + 4x2 f. x2 + 3x + 4 = 0

g. 2x2 + 3x + 1 = 0 h. 2x2 - 5x = 12

342 UNIT 5 • Nonlinear Functions and Equations

5 In addition to expanding and factoring expressions, you can also use a computer algebra system to solve quadratic equations directly. The screen display below shows a CAS solution to the equation 2x2 - 9x - 5 = 0.

Use a calculator or computer algebra system to solve these equations.

a. x2 + 5x + 3 = 0 b. 2x2 - 5x - 12 = 0

c. -10x2 + 240x - 950 = 0 d. x2 - 6x + 10 = 0

The Quadratic Formula Revisited In solving the equations of Problem 5, you saw how some equations that look fairly complex actually have simple whole number solutions, while equations that look quite simple end up with irrational number solutions or even no real number solutions at all.

When quadratic expressions involve fractions or decimals, mental factoring methods are not often easy to use. You will recall from Core-Plus Mathematics Course 1 that for those cases and for situations where you do not have access to a computer algebra system, there is a quadratic formula for finding solutions.

For any equation in the form ax2 + bx + c = 0, the solutions are

x = -b _ 2a +

√ """" b2 - 4ac _ 2a and x = -b

_ 2a - √ """" b2 - 4ac

_ 2a .

To use the quadratic formula in any particular case, all you have to do is

Be sure that the equation is in the form prescribed by the formula.

Identify the values of a, b, and c.

Enter those values where they occur in the formula.

6 Test your understanding and skill with the quadratic formula by using it to solve the following equations. In each case:

• Give the values of a, b, and c that must be used to solve the equations.

• Evaluate -b _ 2a and

√ """" b2 - 4ac _ 2a .

• Evaluate x = -b _ 2a +

√ """" b2 - 4ac _ 2a and x = -b

_ 2a - √ """" b2 - 4ac

_ 2a .

CPMP-Tools

LESSON 1 • Quadratic Functions, Expressions, and Equations 343

• Check that the solutions produced by the formula actually satisfy the equation.

• Graph the related quadratic function to see how the solutions appear as x-intercepts.

a. 5x2 + 3x - 2 = 0 b. x2 - 5x - 6 = 0

c. x2 - 7x + 10 = 0 d. x2 - x - 12 = 0

e. 10 - x2 - 3x = 0 f. 2x2 - 12x + 18 = 0

g. 13 - 6x + x2 = 0 h. -x2 - 4x - 2 = 2

7 Now look back at your work on Problem 6 in search of connections between the quadratic formula calculations and the graphs of the corresponding function rules.

For a quadratic function with rule in the form f(x) = ax2 + bx + c:

a. What information about the graph is provided by -b _ 2a +

√ """" b2 - 4ac _ 2a

and -b _ 2a -

√ """" b2 - 4ac _ 2a ?

b. What information about the graph is provided by the

expression -b _ 2a ?

c. What information about the graph is provided by the

expression √ """" b2 - 4ac

_ 2a ?

Solution Possibilities In solving equations like ax2 + bx + c = 0

with the quadratic formula, the key steps are evaluating -b _ 2a and

√ """" b2 - 4ac _ 2a .

Even if the coefficients a and b and the constant term c are integers, the formula can produce solutions that are not integers.

8 Use the quadratic formula to check each of the following claims about equations and solutions.

a. The solutions of x2 + 5x - 6 = 0 are integers 6 and -1.

b. The solutions of 6x2 + x - 2 = 0 are rational numbers 1 _ 2 and -2 _ 3 .

c. The solutions of x2 - 6x + 4 = 0 are irrational numbers 3 + √ " 5 and 3 - √ " 5 .

d. The equation x2 + 5x + 7 = 0 has no real number solutions.

9 Study the results of your work in Problem 8 to find answers to the questions below.

a. What part of the quadratic formula calculations shows whether there will be 2, 1, or 0 real number solutions? How is that information revealed by the calculations?

b. What part of the quadratic formula calculations shows whether the solutions will be rational or irrational numbers? How is that information revealed by the calculations?