5-Minute Check on Chapter 25-Minute Check on Chapter 2 Transparency 3-1

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

1. Evaluate 42 - |x - 7| if x = -3

2. Find 4.1 (-0.5)

Simplify each expression

3. 8(-2c + 5) + 9c 4. (36d – 18) / (-9)

5. A bag of lollipops has 10 red, 15 green, and 15 yellow lollipops. If one is chosen at random, what is the probability that it is not green?

6. Which of the following is a true statementStandardized Test Practice:

A CB D8/4 < 4/8 -4/8 < -8/4 -4/8 > -8/4 -4/8 > 4/8

Lesson 9-3

Factoring Trinomials: x2 + bx + c

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

Objectives• Factor trinomials of the form x2 + bx + c• Solve equations of the form x2 + bx + c = 0

Vocabulary

• none – means nothing.

Factoring x2 + bx + c

• To factor quadratic trinomials of the form x2 + bx + c, find two integers, m and n, whose sum is equal to b and whose product is equal to c.

• Then write x2 + bx + c using the pattern (x + m)(x + n)

• Symbols: x2 + bx + c = (x + m)(x + n)when m + n = b and m n = c

• Examples: – x2 + 5x + 6 = (x + 2)(x + 3), since 2 + 3 = 5 and 2 3 = 6– x2 + 7x + 10 = (x + 2)(x + 5), since 2 + 5 = 7 and 2 5 = 10– x2 + 9x + 18 = (x + 3)(x + 6), since 3 + 6 = 9 and 3 6 = 18

Multiplication and Division PoEProperties of Equality (PoE) are based on the concept

that as long as you do the same thing to both sides of an equation, then you have not changed anything.

• Multiplication PoE– For any numbers a, b, and c, if a = b, then ac = bc– You can multiply both sides of an equation by the same

thing without changing the equation

• Division PoE– For any numbers a, b, and c with c ≠ 0, if a = b, then a/c = b/c– You can divide both sides of an equation by the same thing

without changing the equation

• Multiplication and division are reciprocal actions

Example 1Factor

In this trinomial, and You need to find the two numbers whose sum is 7 and whose product is 12. Make an organized list of the factors of 12, and look for the pair of factors whose sum is 7.

Factors of 12 Sum of Factors

1, 122, 63, 4

1387 The correct factors are 3 and 4.

Write the pattern.

Answer: and

Example 1 cont

Check You can check the result by multiplying the two factors.

FOIL methodF O I L

Simplify.

Example 2Factor

In this trinomial, and This means is negative and mn is positive. So m and n must both be negative. Therefore, make a list of the negative factors of 27, and look for the pair whose sum is –12.

Factors of 27 Sum of Factors

–1, –27–3, –9

–28–12 The correct factors are –3 and –9.

Write the pattern.

Answer: and

Example 2 contCheck You can check this result by using a graphing calculator. Graph andon the same screen. Since only one graph appears, the two graphs must coincide. Therefore, the trinomial has been factored correctly.

Example 3Factor

In this trinomial, and This means is positive and mn is negative, so either m or n is negative, but not both. Therefore, make a list of the factors of –18 where one factor of each pair is negative. Look for the pair of factors whose sum is 3.

Factors of –18 Sum of Factors 1, –18–1, 18 2, –9–2, 9 3, –6–3, 6

–17 17– 7 7– 3 3 The correct factors are –3 and 6.

Write the pattern.

Answer: and

Example 4

Factors of –20 Sum of Factors

1, –20–1, 20 2, –10–2, 10 4, –5–4, 5

–19 19– 8 8– 1 1 The correct factors are 4 and –5.

FactorSince and is negative and mn is negative. So either m or n is negative, but not both.

Write the pattern.

Answer: and

Example 5Solve

Original equation

Rewrite the equation so that one side equals 0.

Factor.

or Zero Product Property

Solve each equation.

Answer: The solution is

Example 6Architecture Marion has a small art studio measuring 10 feet by 12 feet in her backyard. She wants to build a new studio that has three times the area of the old studio by increasing the length and width by the same amount. What will be the dimensions of the new studio?

Explore Begin by making a diagram like the one shown to the right, labeling the appropriate dimensions.

Example 6 contPlan Let the amount added to each dimension of

the studio.The new length times the new width equals the new area.

old area

Solve Write the equation.

Multiply.

Subtract 360 from each side.

Example 6 cont

Factor.

Solve each equation.

Zero Product Property

or

Examine The solution set is Only 8 is a valid solution, since dimensions cannot be negative.

Answer: The length of the new studio should beor 20 feet and the new width should be or 18 feet.

Summary & Homework

• Summary:– Factoring x2 + bx +c: Find m and n whose

sum is b and whose product is c. – Then write x2 + bx + c as (x + m)(x + n)

• Homework: – Pg. 493. 18-34 even, 38,40,48