1. Find the product. 0.4” (height) Warm-Up Exercises () 8 – m () 9m – ANSWER m 2m 2 17m72 + –z 2z 2 4z4z60 –– ANSWER y 2y 2 400 – ANSWER d 2d 2 18d+81+ ANSWER x 2x 2 28x196 + – ANSWER 1. () 6+z () 10z – 2. () 20+y () y – 3. ()2)2 9+d 4. ()2)2 14x – 5.

2. Warm-Up Exercises 6.A car travels at an average speed of (m 7) miles per hour for (m 2) hours. What distance does it travel? + + ANSWER m 2m 2 9m9m14 ++ () mi

3. Example 1 Factor when c is Positive Factor the expression. x 2x 2 bx+c+ a. x 2x 2 5x5x+6+ b. y 2y 2 6y6y8+ – a.You want (x m) (x n) where mn 6 and m n 5. Because mn is positive, m and n must have the same sign. Since mn 6, find factors of 6 that have a sum of 5. SOLUTION = x 2x 2 5x5x+6+ ++ = + = = Factors of 6 : m, n 7 5 1, 6 7 – – – 2, 3 – – 5 – Sum of factors: m + n

4. Example 1 Factor when c is Positive x 2x 2 bx+c+ ANSWER + + = x 2x 2 5x5x+6+ (x 2) (x 3). Check your answer by multiplying. CHECK () 2+x () 3+x = x 2x 2 3x3x+2x2x+6+ = x 2x 2 5x5x+6+

5. Example 1 Factor when c is Positive x 2x 2 bx+c+ y 2y 2 6y6y8+ – b. You want (y m)(y n) where mn 8 and m n 6. Because mn is positive, m and n must have the same sign. ++ = + = = – Factors of 8 : m, n 9 6 1, 8 9 – – – 2, 4 – – 6 – Sum of factors: m + n ANSWER = (y 2) (y 4). Check your answer by multiplying. –– y 2y 2 6y6y8+ –

6. Example 2 Factor when c is Negative x 2x 2 bx+c+ Factor the expression. a. x 2x 2 8x8x+9 b. z 2z 2 14z15 – – – SOLUTION a.You want (x m) (x n) where mn 9 and m n 8. Because mn is negative, m and n must have different signs. = x 2x 2 8x8x+9 ++ = + = –– Factors of 9 : m, n 0 Sum of factors: m + n – 1, 9 –– 8 3, 3 – 8 – ANSWER (x 1) (x 9). + = x 2x 2 8x8x+9 ––

7. Example 2 Factor when c is Negative x 2x 2 bx+c+ CHECK Check your answer by multiplying. () 1x () 9+x = x 2x 2 9x9x+x9 ––– Multiply using FOIL. = x 2x 2 8x8x+9 – Combine like terms. When you multiply the binomial factors, you obtain the original expression, so the answer is correct.

8. Example 2 Factor when c is Negative x 2x 2 bx+c+ Factors of 15 : m, n Sum of factors: m + n – 1, 15 –– 14 3, 5 – 14 – 2 – – 3, 5 2 ANSWER (z 1) (z 15). Check your answer by multiplying. –z 2z 2 14z15 – = – + b.You want (z m) (z n) where mn 15 and m n 14. Because mn is negative, m and n must have different signs. z 2z 2 14z15 – ++ = + = = – – –

9. Checkpoint Factor the expression. Factor x 2x 2 bx+c+ 1. x 2x 2 6x6x+5+ ANSWER () 1+x () 5+x 2. b 2b 2 7b7b+12+ ANSWER () 3+b () 4+b 3. s 2s 2 5s5s4+ – ANSWER () 4 – s () 1s – () 12+y () 1y – 4. y 2y 2 11y12 + – 5. x 2x 2 x+6 – ANSWER () 3+x () 2x –

10. Checkpoint Factor the expression. Factor x 2x 2 bx+c+ 6. x 2x 2 15x16 –– ANSWER () 16x () 1x+ –

11. Example 3 Solve a Quadratic Equation by Factoring Solve the equation. = x 2x 2 +2x2x15 SOLUTION Write original equation. = x 2x 2 +2x2x15 Write in standard form. = x 2x 2 +2x2x15 – 0 Factor. = 0 () 3x – () 5x + = 03x – or 5x + = 0 Use the zero product property. = 3xx = 5 – Solve for x. ANSWER The solutions are 3 and 5. –

12. Example 4 Use a Quadratic Equation as a Model A group of students from your school volunteers to build a neighborhood playground. The playground will have a mulch border along two sides. The mulch border will have the same width on both sides. The playground is a rectangle, as shown. The length of the playground is 20 yards. The width of the playground is 10 yards. There is enough mulch to cover 64 square yards for the border. How wide should the border be? Community Service

13. Example 4 Use a Quadratic Equation as a Model SOLUTION Use the formula for the area of a rectangle, Area length width. The area of the playground is 20 10 200 square yards. The area of the border will be 64 square yards. So, the total area of the border and the playground will be 264 square yards. = = = 264 x 2x 2 +30x+200 Combine like terms. Multiply using FOIL. = 264 x 2x 2 +10x+20x+200 Formula for area of a rectangle w = A = () 20x + () 10x + 264 Substitute x 20 for and x 10 for w. ++

14. Example 4 Use a Quadratic Equation as a Model = 32x + 0 or 2x – = 0 Use the zero product property. = 32x – 2x = Solve for x. = () 32x + () 2x0 – Factor. Reject 32 as a solution, because a negative width does not make sense. – ANSWER The border should be 2 yards wide. = 0 x 2x 2 +30x64 – Write in standard form.

15. Checkpoint ANSWER 9, 1 Solve the equation. Solve a Quadratic Equation by Factoring ANSWER 7, 2 – 1. = x 2x 2 10x+9 – 0 2. = y 2y 2 5y5y+14 3. = x 2x 2 5 – 4x4x – ANSWER 5, 1 –