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Factoring ax 2 + bx + c by the p-q Method

Factoring ax 2 + bx + c by the p-q Method. Part II: Examples. For each of the following examples, try working one or more steps on paper before proceeding though the slide show to check your work. Factoring ax 2 + bx + c. Find numbers p and q such that

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Factoring ax 2 + bx + c by the p-q Method

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  1. Factoring ax2 + bx + cby the p-q Method Part II: Examples

  2. For each of the following examples, try working one or more steps on paper before proceeding though the slide show to check your work. Factoring ax2 + bx + c

  3. Find numbers p and q such that pq = acand p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 Comparing this with the general form ax2 + bx + c, a = ? b = ? c = ?

  4. Find numbers p and q such that pq = acand p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 Comparing this with the general form ax2 + bx + c, a = 6 b = 11 c= 5

  5. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 Comparing this with the general form ax2 + bx + c, a = 6 b = 11 c = 5 So, pq = ? and p + q = ?

  6. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 Comparing this with the general form ax2 + bx + c, a = 6 b = 11 c = 5 So, pq = 30 and p + q = 11

  7. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 If already know what p and q are, So, pq = 30 and p + q = 11

  8. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 If already know what p and q are, then go on to the next step. Otherwise… So, pq = 30 and p + q = 11

  9. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 Factor 1, 2, 3,… out of 30, until you find two numbers whose sum is 11: So, pq = 30 and p + q = 11 130 = 30 1 + 30 = 31 215 = 30 2 + 15 = 17 310 = 30 3 + 10 = 13 56 = 30 5 + 6 = 11 So, p = 5 and q = 6 (or vise versa).

  10. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 6x2 + 11x+ 5 = 6x2 + 5x + 6x + 5 So, pq = 30 and p + q = 11 130 = 30 1 + 30 = 31 215 = 30 2 + 15 = 17 310 = 30 3 + 10 = 13 56 = 30 5 + 6 = 11 So, p = 5 and q = 6 (or vise versa).

  11. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 6x2 + 11x + 5 = 6x2 + 5x + 6x + 5 So, pq = 30 and p + q = 11 130 = 30 1 + 30 = 31 215 = 30 2 + 15 = 17 310 = 30 3 + 10 = 13 56 = 30 5 + 6 = 11 So, p = 5 and q = 6 (or vise versa).

  12. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 6x2 + 11x + 5 = 6x2 + 5x + 6x + 5 = (6x2 + 5x) + (6x + 5) So, pq = 30 and p + q = 11 130 = 30 1 + 30 = 31 215 = 30 2 + 15 = 17 310 = 30 3 + 10 = 13 56 = 30 5 + 6 = 11 So, p = 5 and q = 6 (or vise versa).

  13. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 6x2 + 11x + 5 = 6x2 + 5x + 6x + 5 = (6x2 + 5x) + (6x + 5) = x(6x + 5) + 1(6x + 5) So, pq = 30 and p + q = 11 130 = 30 1 + 30 = 31 215 = 30 2 + 15 = 17 310 = 30 3 + 10 = 13 56 = 30 5 + 6 = 11 So, p = 5 and q = 6 (or vise versa).

  14. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 6x2 + 11x + 5 = 6x2 + 5x + 6x + 5 = (6x2 + 5x) + (6x + 5) = x(6x + 5) + 1(6x + 5) = (6x + 5)(x +1) So, pq = 30 and p + q = 11 130 = 30 1 + 30 = 31 215 = 30 2 + 15 = 17 310 = 30 3 + 10 = 13 56 = 30 5 + 6 = 11 So, p = 5 and q = 6 (or vise versa).

  15. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 1: Factor 6x2 + 11x + 5 6x2 + 11x + 5 = 6x2 + 5x + 6x + 5 = (6x2 + 5x) + (6x + 5) = x(6x + 5) + 1(6x + 5) = (6x + 5)(x +1) That’s it. So, pq = 30 and p + q = 11 130 = 30 1 + 30 = 31 215 = 30 2 + 15 = 17 310 = 30 3 + 10 = 13 56 = 30 5 + 6 = 11 So, p = 5 and q = 6 (or vise versa).

  16. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10

  17. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 Comparing with the general “quadratic” form ax2 + bx + c, a = 4 b = 3 c = 10 It’s essential that the sign, + or , of these coefficients be properly accounted for.

  18. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 Comparing with the general “quadratic” form ax2 + bx + c, a = 4 b = 3 c = 10 It’s essential that the sign, + or , of these coefficients be properly accounted for. So, pq = 40 and p + q = 3

  19. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 pq = 40 • p or q is negative, but not both. So, pq = 40 and p + q = 3

  20. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 pq = 40 • p or q is negative, but not both. p + q = 3 > 0  The positive one has the larger absolute value. So, pq = 40 and p + q = 3

  21. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 pq = 40 • p or q is negative, but not both. p + q = 3 > 0 • The positive one has the larger absolute value. Therefore, we’ll be factoring out 1, 2, etc. instead of 1, 2,… So, pq = 40 and p + q = 3

  22. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 pq = 40 • p or q is negative, but not both. p + q = 3 > 0 • The positive one has the larger absolute value. Therefore, we’ll be factoring out 1, 2, etc. instead of 1, 2,… So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3

  23. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 pq = 40 • p or q is negative, but not both. p + q = 3 > 0 • The positive one has the larger absolute value. Therefore, we’ll be factoring out 1, 2, etc. instead of 1, 2,… So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3 So, p = 5 and q = 8

  24. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3 So, p = 5 and q = 8

  25. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 4x2 + 3x 10 = 4x25x + 8x 8 So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3 So, p = 5 and q = 8

  26. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 4x2 + 3x 10 = 4x25x + 8x 8 So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3 So, p = 5 and q = 8

  27. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 4x2 + 3x 10 = 4x25x + 8x 10 = (4x25x) + (8x 10) So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3 So, p = 5 and q = 8

  28. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 4x2 + 3x 10 = 4x25x + 8x 10 = (4x25x) + (8x 10) = x(4x – 5) + 2(4x – 5) So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3 So, p = 5 and q = 8

  29. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 4x2 + 3x 10 = 4x25x + 8x 10 = (4x25x) + (8x 10) = x(4x – 5) + 2(4x – 5) = (4x – 5)(x + 2) So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3 So, p = 5 and q = 8

  30. Find numbers p and q such that pq = ac and p + q = b. Split the middle term into two terms: ax2 + bx + c = ax2 + px + qx + c Factor by grouping. Example 2: Factor 4x2 + 3x10 4x2 + 3x 10 = 4x25x + 8x 10 = (4x25x) + (8x 10) = x(4x – 5) + 2(4x – 5) = (4x – 5)(x + 2) So, pq = 40 and p + q = 3 (1)40 = 40, 1 + 40 = 39 (2)20 = 40, 2 + 20 = 18 (4)10 = 40, 4 + 10 = 8 (5)8 = 40, 5 + 8 = 3 So, p = 5 and q = 8

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