For Educational Use Only © 2010 10.6 Factoring ax 2 + bx + c Brian Preston Algebra 1 2009-2010.

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For Educational Use Only © 2010 10.6 Factoring ax 2 + bx + c Brian Preston Algebra 1 2009-2010

For Educational Use Only © 2010 shirt Real World Application How long until you catch the shirt? 70 ft. 6 ft.

For Educational Use Only © 2010 Lesson Objectives 1) Factor a quadratic expression of the form ax 2 + bx + c. 2) Solve quadratic equations by factoring.

For Educational Use Only © 2010 1) x 2 – 2x – 8 1 Factors of 8 1  8 – 2 -2 = 2  4 Factor the trinomial. Review 1 + 8 9 = 8 – 1 7 =

For Educational Use Only © 2010 1) x 2 – 2x – 8 1 2 + – 4 -2 = Factor the trinomial. 1  8 2  4 Factors of 8 Review 2 + 4 6 = 4 – 2 2 =  (–1) to all

For Educational Use Only © 2010 1) x 2 – 2x – 8 – 4 1x 1 +2 2 + – 4 – 4 2 ( + 2 -2 = Factor the trinomial. 1  8 2  4 Factors of 8 )( 1x ) 1x 2 1x Review

For Educational Use Only © 2010 2) A mascot shoots t-shirts from a “T-shirt cannon” into the crowd. The T-shirts are released from a height of 6 feet with an initial upward velocity of 72 feet per second. If you catch a T-shirt at your seat 70 feet above the court, how long was it in the air before you caught it? 72 70 + = – 16t 2 long 6 6 Real World Application 72t hvs 706 upward +

For Educational Use Only © 2010 – 70 – 4 Real World Application 2) – 16t 2 + 72t + 6 = 70 Solve the equation by factoring. – 70 = + 72t – 16t 2 0 – 64 Standard form = – 18t 4t 2 + 16 0 – 4

For Educational Use Only © 2010 – 18 444 16 a ax 2 + bx + c = 0 (4) (16) (– 18) 2) 4t 2 – 18t + 16 = 0 a = b = – 18 4 c = 16 t = b 2 – 4ac – b ± 2a t = 2 – 42 – 4 – ± 2 (– 18) b b c a Standard form Real World Application

For Educational Use Only © 2010 (4) 2) 324 t = – 256 ± 8 18 68 t = ± 8 18 (4) (16) (–18) t = 2 – 42 – 4 – ± 2 (–18) Real World Application

For Educational Use Only © 2010 t = t =t = 8.24 ± 8 18 8.24 + 8 18 = 8.24 – 8 18 = 3.28 1.22 t = 3.28 seconds 2) Real World Application Way down Way up

For Educational Use Only © 2010 shirt Real World Application How long until you catch the shirt? 70 ft. 6 ft. 3.28 seconds

For Educational Use Only © 2010 Example 3) 3x 2 – 4x - 7 Factors of 21 1  21 – 4 -4 = 3  7 Now the numbers are getting bigger. 1 + 21 22 = 21 – 1 20 =

For Educational Use Only © 2010 3 + – 7 Example 3) 3x 2 – 4x - 7 -4 = Factor the trinomial. 1  21 3  7 Factors of 21 3 + 7 10 = 7 – 3 4 =  (–1) to all

For Educational Use Only © 2010 3 + – 7 – 7 3 ( + 1 Example 3) 3x 2 – 4x - 7 -4 = Factor the trinomial. 1  21 3  7 Factors of 21 +3 – 7 3x )( 1x ) 3x 2 3x +3 – 7 1x

For Educational Use Only © 2010 Rule x 2 – 2x – 8 One (-) & One (+) Patterns for factoring trinomials. (x + 2) (x - 4) x 2 + 6x + 8 (x + 2) (x + 4) x 2 – 6x + 8 (x – 2) (x – 4) Two (+) Two (-) x 2 + 2x – 8 (x + 4) (x - 2)

For Educational Use Only © 2010 1 + 10 Example 4) 2x 2 + 11x + 5 Factors of 10 1  10 + 11 = 2  5 Factor the trinomial. 1 + 10 11 = 10 – 1 9 =

For Educational Use Only © 2010 +1 +10 2x 1 + 10 10 4) 2x 2 + 11x + 5 1 ( Example +11 = Factor the trinomial. 1  10 2  5 Factors of 10 )( 1x ) 2x 2 Switch

For Educational Use Only © 2010 +10 +1 2x 1 + 10 4) 2x 2 + 11x + 5 ( + 1 + 5 Example +11 = Factor the trinomial. 1  10 2  5 Factors of 10 )( 1x )

For Educational Use Only © 2010 Example 5) 6a 2 – 19a + 15 Factors of 90 1  90 – 19 -19 = 2  45 Factor the trinomial. 3  30 5  18 6  15 9  10 1 + 90 91 = 90 – 1 89 =

For Educational Use Only © 2010 Example 5) 6a 2 – 19a + 15 Factors of 90 1  90 -19 = 2  45 Factor the trinomial. 3  30 5  18 6  15 9  10 2 + 45 47 = 45 – 2 43 =

For Educational Use Only © 2010 Example 5) 6a 2 – 19a + 15 Factors of 90 1  90 -19 = 2  45 Factor the trinomial. 3  30 5  18 6  15 9  10 3 + 30 33 = 30 – 3 27 =

For Educational Use Only © 2010 Example 5) 6a 2 – 19a + 15 Factors of 90 1  90 -19 = 2  45 Factor the trinomial. 3  30 5  18 6  15 9  10 5 + 18 23 = 18 – 5 13 =

For Educational Use Only © 2010 Example 5) 6a 2 – 19a + 15 Factors of 90 1  90 -19 = 2  45 Factor the trinomial. 3  30 5  18 6  15 9  10 6 + 15 21 = 15 – 6 9 =

For Educational Use Only © 2010 5) 6a 2 – 19a + 15 – 9 + – 10 Example -19 = Factor the trinomial. Factors of 90 1  90 2  45 3  30 5  18 6  15 9  10 9 + 10 19 = 10 – 9 1 =  (–1) to all

For Educational Use Only © 2010 -9 –10 -9 –10 2a 3a 5) 6a 2 – 19a + 15 – 9 + – 10 – 10 – 9 ( – 5 – 3 Example -19 = Factor the trinomial. Factors of 90 )( ) 6a 2 1  90 2  45 3  30 5  18 6  15 9  10

For Educational Use Only © 2010 Example 6) 6x 2 – 2x – 8 Factors of 48 1  48 – 2 -2 = 2  24 Factor the trinomial. 3  16 4  12 6  8 1 + 48 49 = 48 – 1 47 =

For Educational Use Only © 2010 6) 6x 2 – 2x – 8 Example Factors of 48 1  48 -2 = 2  24 Factor the trinomial. 3  16 4  12 6  8 2 + 24 26 = 24 – 2 22 =

For Educational Use Only © 2010 6) 6x 2 – 2x – 8 Example Factors of 48 1  48 -2 = 2  24 Factor the trinomial. 3  16 4  12 6  8 3 + 16 19 = 16 – 3 13 =

For Educational Use Only © 2010 6) 6x 2 – 2x – 8 Example Factors of 48 1  48 -2 = 2  24 Factor the trinomial. 3  16 4  12 6  8 4 + 12 16 = 12 – 4 8 =

For Educational Use Only © 2010 6) 6x 2 – 2x – 8 6 + – 8 Example -2 = Factor the trinomial. Factors of 48 1  48 2  24 3  16 4  12 6  8 6 + 8 14 = 8 – 6 2 =  (–1) to all

For Educational Use Only © 2010 +6 –8 +6 –8 3x 2x 6) 6x 2 – 2x – 8 6 + – 8 – 8 6 ( – 4 + 2 Example -2 = Factor the trinomial. Factors of 48 )( ) 6x 2 1  48 2  24 3  16 4  12 6  8

For Educational Use Only © 2010 Example 7) 8t 2 – 4t – 5 Factors of 40 1  40 – 4 -4 = 2  20 Factor the trinomial. 4  10 5  8 1 + 40 41 = 40 – 1 39 =

For Educational Use Only © 2010 7) 8t 2 – 4t – 5 Example Factors of 40 1  40 -4 = 2  20 Factor the trinomial. 4  10 5  8 2 + 20 22 = 20 – 2 18 =

For Educational Use Only © 2010 7) 8t 2 – 4t – 5 Example Factors of 40 1  40 -4 = 2  20 Factor the trinomial. 4  10 5  8 4 + 10 14 = 10 – 4 6 =

For Educational Use Only © 2010 7) 8t 2 – 4t – 5 Example Factors of 40 1  40 -4 = 2  20 Factor the trinomial. 4  10 5  8 Not Factorable 5 + 8 13 = 8 – 5 3 =

For Educational Use Only © 2010 Example 8) 2z 2 – 17z – 19 = 0 Factors of 38 1  38 – 17 -17 = 2  19 Solve the equation by factoring. 1 + 38 39 = 38 – 1 37 =

For Educational Use Only © 2010 8) 2z 2 – 17z – 19 = 0 2 + – 19 Example -17 = Solve the equation by factoring. 1  38 2  19 Factors of 38 2 + 19 21 = 19 – 2 17 =  (–1) to all

For Educational Use Only © 2010 2z 1z2z1z +2 –19 8) 2z 2 – 17z – 19 = 0 2 + – 19 – 19 2 ( + 1 Example -17 = Solve the equation by factoring. 1  38 2  19 Factors of 38 )( )=0 2z 2

For Educational Use Only © 2010 2 + 19 2z – 19 + 19 2z 2 – 1 (2z – 19) (1z + 1) (2z – 19) (1z + 1) Example Solve the equation by factoring. 8) = 0 1z + 1 + 19 = 19 ( ) ( ) z – 1 = + 0 2 z= z=– 1 19 2

For Educational Use Only © 2010 + 4x – 2 Example 9) 28x 2 – 9x – 1 = – 4x + 2 Solve the equation by factoring. – 2 = – 9x 28x 2 + 0 – 3 Standard form – 4x + 4x = – 5x 28x 2 – 3 0

For Educational Use Only © 2010 Example 9) 28x 2 – 5x – 3 = 0 Factors of 84 1  84 – 5 -5 = 2  42 Solve the equation by factoring. 3  28 4  21 6  14 7  12 1 + 84 85 = 84 – 1 83 =

For Educational Use Only © 2010 Example 9) 28x 2 – 5x – 3 = 0 Factors of 84 1  84 -5 = 2  42 Solve the equation by factoring. 3  28 4  21 6  14 7  12 2 + 42 44 = 42 – 2 40 =

For Educational Use Only © 2010 Example 9) 28x 2 – 5x – 3 = 0 Factors of 84 1  84 -5 = 2  42 Solve the equation by factoring. 3  28 4  21 6  14 7  12 3 + 28 31 = 28 – 3 25 =

For Educational Use Only © 2010 Example 9) 28x 2 – 5x – 3 = 0 Factors of 84 1  84 -5 = 2  42 Solve the equation by factoring. 3  28 4  21 6  14 7  12 4 + 21 25 = 21 – 4 17 =

For Educational Use Only © 2010 Example 9) 28x 2 – 5x – 3 = 0 Factors of 84 1  84 -5 = 2  42 Solve the equation by factoring. 3  28 4  21 6  14 7  12 6 + 14 20 = 14 – 6 8 =

For Educational Use Only © 2010 9) 28x 2 – 5x – 3 = 0 7 + – 12 Example -5 = Solve the equation by factoring. Factors of 84 1  84 2  42 3  28 4  21 6  14 7  12 7 + 12 19 = 12 – 7 5 =  (–1) to all

For Educational Use Only © 2010 4x 7x +7 –12 +7 –12 9) 28x 2 – 5x – 3 = 0 7 + – 12 – 12 7 ( – 3 + 1 Example -5 = Solve the equation by factoring. Factors of 84 )( )=0 28x 2 1  84 2  42 3  28 4  21 6  14 7  12

For Educational Use Only © 2010 4 4 + 3 7 (4x + 1) (7x – 3) (4x + 1) 7x – 3 7x 7 – 1 Example Solve the equation by factoring. 9) = 0 4x + 1 + 3 = 3 ( ) ( ) 4x – 1 = + 0 7 x= x= 3 7 1 4 – 4

For Educational Use Only © 2010 1) Don’t forget the negative signs. 2) Make sure the expressions and equations are in standard form before factoring. Key Points & Don’t Forget

For Educational Use Only © 2010 pg. 509-510 #’s 16-44 even The Assignment

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