1

You want (x m) (x n) where mn 6 Example 1 Factor when c is Positive x 2 bx + c Factor the expression. a. b. y 2 6y 8 + – x 2 + 5x + 6 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 2 5x + 6 Factors of 6: m, n 7 5 1, 6 – 1, 6 2, 3 2, 3 Sum of factors: m + n 2

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

(y 2) (y 4). Check your answer by multiplying. – y 2 6y 8 + Example 1 Factor when c is Positive x 2 bx + c y 2 6y 8 + – 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 – 1, 8 2, 4 2, 4 Sum of factors: m + n ANSWER = (y 2) (y 4). Check your answer by multiplying. – y 2 6y 8 + 4

You want (x m) (x n) where mn 9 Example 2 Factor when c is Negative x 2 bx + c Factor the expression. a. – b. x 2 + 8x 9 z 2 – 14z – 15 SOLUTION 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 2 8x + 9 – Factors of 9: m, n Sum of factors: m + n 1, 9 1, 9 8 3, 3 ANSWER (x 1) (x 9). + = x 2 8x 9 – 5

CHECK Check your answer by multiplying. Example 2 Factor when c is Negative x 2 bx + c CHECK Check your answer by multiplying. ( ) 1 x 9 + = x 2 9x – Multiply using FOIL. = x 2 8x + 9 – Combine like terms. When you multiply the binomial factors, you obtain the original expression, so the answer is correct. 6

You want (z m) (z n) where mn 15 and m n 14. Because mn is negative, Example 2 Factor when c is Negative x 2 bx + c 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 2 14z 15 – + = Factors of 15: m, n Sum of factors: m + n – 1, 15 1, 15 14 3, 5 2 3, 5 ANSWER (z 1) (z 15). Check your answer by multiplying. – z 2 14z 15 = + 7

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

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

10

Example 3 Solve the equation . = x 2 + 2x 15 SOLUTION = x 2 + 2x 15 = Solve a Quadratic Equation by Factoring Solve the equation . = x 2 + 2x 15 SOLUTION Write original equation. = x 2 + 2x 15 Write in standard form. = x 2 + 2x 15 – Factor. = ( ) 3 x – 5 + = 3 x – or 5 + Use the zero product property. = 3 x 5 – Solve for x. ANSWER The solutions are 3 and 5. –

A group of students from your school Example 4 Use a Quadratic Equation as a Model Community Service 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? 12

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. = • Formula for area of a rectangle w = • A = ( ) 20 x + 10 264 Substitute x 20 for and x 10 for w. Multiply using FOIL. = 264 x 2 + 10x 20x 200 = 264 x 2 + 30x 200 Combine like terms. 13

Reject 32 as a solution, because a negative width does not make sense. Example 4 Use a Quadratic Equation as a Model = x 2 + 30x 64 – Write in standard form. = ( ) 32 x + 2 – Factor. = 32 x + or 2 – Use the zero product property. = 32 x – 2 Solve for x. Reject 32 as a solution, because a negative width does not make sense. – ANSWER The border should be 2 yards wide. 14

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

Your school plans to increase the area of the parking Parking Lot Your school plans to increase the area of the parking lot by 1000 square yards. The original parking lot is a rectangle, as shown. The length and width of the parking lot will each increase by x yards. The width of the original parking lot is 40 yards, and the length of the original parking lot is 50 yards. 1) Find the area of the original parking lot. 2) Find the total area of the parking lot with the new space. 3) Write an equation that you can use to find the value of x. 16

17

18

19

20