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For Educational Use Only © 2010 10.5 Factoring x 2 + bx + c Brian Preston Algebra 1 2009-2010

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For Educational Use Only © 2010 Real World Application How wide is the stone border?

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

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For Educational Use Only © 2010 Now we are going to learn how to do this backwards. Review (x + 4)(x + 5)5 x5x 4 4 x x 1) x2x2 + 4x + 5x+ 20 x2x2 + 9x + 20 First Outside Inside Last

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For Educational Use Only © 2010 2) x 2 + 3x + 2 1 1 + 2 Factors of 2 1 2 + 3 = Factor the trinomial. Example

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For Educational Use Only © 2010 +2 +1 +2 +1 2) x 2 + 3x + 2 1x 1 1 + 2 2 1 ( + 2 + 1 +3 = Factor the trinomial. 1 2 Factors of 2 )( 1x ) 1x 2 1x Example

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For Educational Use Only © 2010 3) Mike is building a stone border along two sides of a rectangular Japanese garden that measures 6 yards by 15 yards. His budget limits him to only enough stone to cover 46 square yards. How wide should the border be? border 46 6 15 (15)(6) Are of border 46(x + 6) 46 6 Real World Application = Garden area – (x + 15) Total area 15 border 6 15

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For Educational Use Only © 2010 3)(x + 15)(x + 6) – (15)(6) = 46 6156 – 46 Real World Application – 46 = + 21x x2x2 0 – 46 x2x2 + 15x + 6x+ 90 x2x2 + 21x + 0 – 90 = 46 x x xx

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For Educational Use Only © 2010 3) x 2 + 21x – 46 = 0 1 Factors of 46 1 46 + 21 = 2 23 Factor the trinomial. Real World Application 1 + 46 47 = 46 – 1 45 =

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For Educational Use Only © 2010 3) x 2 + 21x – 46 = 0 1 – 2 + 23 +21 = Factor the trinomial. 1 46 2 23 Factors of 46 Real World Application 2 + 23 25 = 23 – 2 21 =

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For Educational Use Only © 2010 3) x 2 + 21x – 46 = 0 +23 -2 +23 -2 1 1x 2 1x – 2 + 23 23 – 2 ( + 23 – 2 +21 = Factor the trinomial. 1 46 2 23 Factors of 46 )( ) Real World Application = 0= 0

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For Educational Use Only © 2010 – 23 + 2 (x + 23) (x – 2) (x + 23) (x – 2) x + 23 Example Solve the equation by factoring. 3) = 0 x – 2 x – 23 = ( ) ( ) x + 2 = 2 2 yards wide

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For Educational Use Only © 2010 Real World Application How wide is the stone border? 2 yards wide

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For Educational Use Only © 2010 What is factoring? Factoring is another way to solve for variables in a quadratic equations.

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For Educational Use Only © 2010 4) x 2 – 5x + 6 1 1 + – 6 Factors of 6 1 6 – 5 -5 = 2 3 Factor the trinomial. Example 1 + 6 7 = 6 – 1 5 = (–1) to all 1 + – 6

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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)

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For Educational Use Only © 2010 4) x 2 – 5x + 6 1 – 2 + – 3 -5 = Factor the trinomial. 1 6 2 3 Factors of 6 Example 2 + 3 5 = 3 – 2 1 = (–1) to all

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For Educational Use Only © 2010 – 3 -2 – 3 -2 4) x 2 – 5x + 6 1x 1 – 2 + – 3 – 3 – 2 ( – 3 – 2 -5 = Factor the trinomial. 1 6 2 3 Factors of 6 )( 1x ) 1x 2 1x Example

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For Educational Use Only © 2010 5) x 2 – 2x – 8 1 Factors of 8 1 8 – 2 -2 = 2 4 Factor the trinomial. Example 1 + 8 9 = 8 – 1 7 =

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For Educational Use Only © 2010 5) x 2 – 2x – 8 1 2 + – 4 -2 = Factor the trinomial. 1 8 2 4 Factors of 8 Example 2 + 4 6 = 4 – 2 2 = (–1) to all

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For Educational Use Only © 2010 5) 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 Example

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For Educational Use Only © 2010 1 6) x 2 + 7x – 18 Factors of 18 1 18 + 7 = 2 9 Factor the trinomial. Example 3 6 1 + 18 19 = 18 – 1 17 =

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For Educational Use Only © 2010 6) x 2 + 7x – 18 1 – 2 + 9 +7 = Factor the trinomial. 1 18 2 9 Factors of 18 Example 3 6 2 + 9 11 = 9 – 2 7 =

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For Educational Use Only © 2010 +9 -2 +9 -2 6) x 2 + 7x – 18 1x 1 – 2 + 9 9 – 2 ( + 9 – 2 +7 = Factor the trinomial. 1 18 2 9 Factors of 18 )( 1x ) 1x 2 1x Example 3 6

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For Educational Use Only © 2010 1 7) w 2 + 13w + 36 Factors of 36 + 13 = Factor the trinomial. Example 1 36 2 18 3 12 4 9 6 6 1 + 36 37 = 36 – 1 35 =

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For Educational Use Only © 2010 7) w 2 + 13w + 36 1 Factors of 36 +13 = Factor the trinomial. Example 1 36 2 18 3 12 4 9 6 6 2 + 18 20 = 18 – 2 16 =

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For Educational Use Only © 2010 7) w 2 + 13w + 36 1 Factors of 36 +13 = Factor the trinomial. Example 1 36 2 18 3 12 4 9 6 6 3 + 12 15 = 12 – 3 9 =

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For Educational Use Only © 2010 7) w 2 + 13w + 36 1 4 + 9 +13 = Factor the trinomial. Factors of 36 Example 1 36 2 18 3 12 4 9 6 6 4 + 9 13 = 9 – 4 5 =

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For Educational Use Only © 2010 7) w 2 + 13w + 36 +4 1w +9 1 4 + 9 9 4 ( + 9 + 4 +13 = Factor the trinomial. Factors of 36 )( ) 1w 2 Example 1 36 2 18 3 12 4 9 6 6

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For Educational Use Only © 2010 8) 32 + 12n + n 2 (4 + n) (3 + n) (n + 1) (n + 2) Example Factor the trinomial.

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For Educational Use Only © 2010 8) 32 + 12n + n 2 Example Factors of 32 1 32 + 12 = 2 16 Factor the trinomial. 1 3 12 4 8 6 6 1 + 32 33 = 32 – 1 31 =

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For Educational Use Only © 2010 8) 32 + 12n + n 2 +12 = Factor the trinomial. Example 1 36 2 18 3 12 4 8 6 6 Factors of 32 1 2 + 18 20 = 18 – 2 16 =

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For Educational Use Only © 2010 8) 32 + 12n + n 2 +12 = Factor the trinomial. Example 1 36 2 18 3 12 4 8 6 6 1 Factors of 32 3 + 12 15 = 12 – 3 9 =

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For Educational Use Only © 2010 8) 32 + 12n + n 2 1 4 + 8 +12 = Factor the trinomial. Example 1 36 2 18 3 12 4 8 6 6 Factors of 32 4 + 8 12 = 8 – 4 4 =

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For Educational Use Only © 2010 +8 8 + 8) 32 + 12n + n 2 1n +4 1 4 + 8 8 4 ( 4 + +12 = Factor the trinomial. )( ) Example 1 36 2 18 3 12 4 8 6 6 Factors of 32 1n 2

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For Educational Use Only © 2010 Example 3 3 x 2 + 3x – 4 – 4 (1) (– 4) a (3) 9) Tell whether the trinomial can be factored. 1 ax 2 + bx + c = 0 a = b = 3 1 1 c = – 4 b 2 – 4ac 2 – 4 b c 1 Standard form

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For Educational Use Only © 2010 (1) (– 4) (3) 9) 2 – 4 9 + 16 25 25 is a perfect square, so Yes Example

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For Educational Use Only © 2010 – 6 3 3 x 2 + 3x – 6 (1) (– 6) a (3) 10) Tell whether the trinomial can be factored. 1 ax 2 + bx + c = 0 a = b = 3 1 1 c = – 6 b 2 – 4ac 2 – 4 b c 1 Standard form Example

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For Educational Use Only © 2010 (1) (– 6) (3) 10) 2 – 4 9 + 24 33 33 is not a perfect square, so No Example

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For Educational Use Only © 2010 11) b 2 + 7b + 10 = 0 1 Factors of 10 1 10 + 7 = 2 5 Solve the equation by factoring. Example Standard form 1 + 10 11 = 10 – 1 9 =

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For Educational Use Only © 2010 11) b 2 + 7b + 10 = 0 1 2 + 5 +7 = Solve the equation by factoring. 1 10 2 5 Factors of 10 Example 2 + 5 7 = 5 – 2 3 =

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For Educational Use Only © 2010 +5 1b+ 5 + 2 1b 11) b 2 + 7b + 10 = 0 1 +2 2 + 5 5 2 ( +7 = Solve the equation by factoring. 1 10 2 5 Factors of 10 )( ) 1b 2 Example = 0

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For Educational Use Only © 2010 – 5 – 2– 5 – 2 (b + 5) (b + 2) (b + 5) (b + 2) b + 5 Example Solve the equation by factoring. 11) = 0 b + 2 b – 5 = ( ) ( ) b – 2 =

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For Educational Use Only © 2010 12) x 2 + 5x – 14 = 0 1 Factors of 14 1 14 + 5 = 2 7 Solve the equation by factoring. Example Standard form 1 + 14 15 = 14 – 1 13 =

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For Educational Use Only © 2010 12) x 2 + 5x – 14 = 0 1 – 2 + 7 +5 = Solve the equation by factoring. 1 14 2 7 Factors of 14 Example 2 + 7 9 = 7 – 2 5 =

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For Educational Use Only © 2010 +7 -2 +7 -2 1x 12) x 2 + 5x – 14 = 0 + 7 – 2 1 – 2 + 7 7 – 2 ( +5 = Solve the equation by factoring. 1 14 2 7 Factors of 14 )( ) 1x 2 Example = 0

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For Educational Use Only © 2010 – 7 + 2– 7 + 2 (x + 7) (x – 2) (x + 7) (x – 2) x + 7 Example Solve the equation by factoring. 12) = 0 x – 2 x – 7 = ( ) ( ) x + 2 = 2

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For Educational Use Only © 2010 + 4 Example 13) x 2 – 2x – 19 = – 4 Solve the equation by factoring. + 4 = – 2x x2x2 0 – 15 Standard form

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For Educational Use Only © 2010 13) x 2 – 2x – 15 = 0 1 Factors of 15 1 15 – 2 -2 = 3 5 Solve the equation by factoring. Example 1 + 15 16 = 15 – 1 14 =

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For Educational Use Only © 2010 13) x 2 – 2x – 15 = 0 1 3 + – 5 -2 = Solve the equation by factoring. 1 15 3 5 Factors of 15 Example 3 + 5 8 = 5 – 3 2 = (–1) to all

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For Educational Use Only © 2010 -5 +3 13) x 2 – 2x – 15 = 0 1x – 5 + 3 1 3 + – 5 – 5 3 ( -2 = Solve the equation by factoring. 1 15 3 5 Factors of 15 )( ) 1x 2 Example = 0

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For Educational Use Only © 2010 + 5 – 3+ 5 – 3 (x – 5) (x + 3) x – 5 Example Solve the equation by factoring. 13) = 0 x + 3 x + 5 = 5 ( ) ( ) x – 3 = (x – 5)

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For Educational Use Only © 2010 + 15 Example 14) x 2 + 16x = – 15 Solve the equation by factoring. = + 16x x2x2 0 + 15 Standard form

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For Educational Use Only © 2010 14) x 2 + 16x + 15 = 0 1 1 + 15 Factors of 15 1 15 + 16 = 3 5 Solve the equation by factoring. Example 1 + 15 16 = 15 – 1 14 =

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For Educational Use Only © 2010 +1 +15 ( 1 + 15 14) x 2 + 16x + 15 = 0 1x + 15 + 1 1 15 1 ( +16 = Solve the equation by factoring. 1 15 3 5 Factors of 15 ) ) 1x 2 Example = 0

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For Educational Use Only © 2010 – 1 – 15 (x + 15) (x + 1) (x + 15) (x + 1) x + 15 Example Solve the equation by factoring. 14) = 0 x + 1 x – 15 = ( ) ( ) x – 1 =

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

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For Educational Use Only © 2010 pg. 462-463 #’s 10-37, 40-46 even The Assignment

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For Educational Use Only © 2010 Please email brianspowerpoints@gmail.com with errors, confusing slides, improvements, complications, or any other comments or questions.brianspowerpoints@gmail.com The template is from www.spiralgraphics.bizwww.spiralgraphics.biz http://www.worldofteaching.comhttp://www.worldofteaching.com is home to over a thousand powerpoints submitted by teachers. This is a completely free site and requires no registration. Please visit and I hope it will help in your teaching. Bibliography

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