5 Minute Check on Chapter 2 Transparency 3

5 -Minute Check on Chapter 2 Transparency 3 -1 1. Evaluate 42 - |x - 7| if x = -3 2. Find 4. 1 (-0. 5) Simplify each expression 3. 8(-2 c + 5) + 9 c 4. (36 d – 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. Standardized Test Practice: A 8/4 < 4/8 B Which of the following is a true statement -4/8 < -8/4 C -4/8 > -8/4 Click the mouse button or press the Space Bar to display the answers. D -4/8 > 4/8

Lesson 9 -3 Factoring Trinomials: x 2 + bx + c

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

Objectives • Factor trinomials of the form x 2 + bx + c • Solve equations of the form x 2 + bx + c = 0

Vocabulary • none – means nothing.

Factoring x 2 + bx + c • To factor quadratic trinomials of the form x 2 + bx + c, find two integers, m and n, whose sum is equal to b and whose product is equal to c. • Then write x 2 + bx + c using the pattern (x + m)(x + n) • Symbols: x 2 + bx + c = (x + m)(x + n) when m + n = b and m n = c • Examples: – x 2 + 5 x + 6 = (x + 2)(x + 3), since 2 + 3 = 5 and 2 3 = 6 – x 2 + 7 x + 10 = (x + 2)(x + 5), since 2 + 5 = 7 and 2 5 = 10 – x 2 + 9 x + 18 = (x + 3)(x + 6), since 3 + 6 = 9 and 3 6 = 18

Multiplication and Division Po. E Properties of Equality (Po. E) 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 Po. E – 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 Po. E – 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 1 Factor 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, 12 2, 6 3, 4 13 8 7 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. F O I L FOIL method Simplify.

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

Example 3 Factor 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 1, – 18 – 1, 18 2, – 9 – 2, 9 3, – 6 – 3, 6 Sum of Factors – 17 17 – 7 7 – 3 The correct factors are – 3 and 6. 3 Write the pattern. Answer: and

Example 4 Factor Since and is negative and mn is negative. So either m or n is negative, but not both. 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. Write the pattern. Answer: and

Example 5 Solve Original equation Rewrite the equation so that one side equals 0. Factor. or Zero Product Property Solve each equation. Answer: The solution is

Example 6 Architecture 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 cont Plan 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. or Zero Product Property Solve each equation. Examine The solution set is Only 8 is a valid solution, since dimensions cannot be negative. Answer: The length of the new studio should be or 20 feet and the new width should be or 18 feet.

Summary & Homework • Summary: – Factoring x 2 + bx +c: Find m and n whose sum is b and whose product is c. – Then write x 2 + bx + c as (x + m)(x + n) • Homework: – Pg. 493. 18 -34 even, 38, 40, 48