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Factoring ax2 + bx + c

This section explains how to factor expressions of the form ax2 + bx + c, where a, b, and c are integers.

First, factor out all constants which evenly divide all three terms. If a is negative, factor out -1. This will leave an expression of the form d (ax2 + bx + c), where a, b, c, and d are integers, and a > 0. We can now turn to factoring the inside expression.

Here is how to factor an expression ax2 + bx + c, where a > 0:

  1. Write out all the pairs of numbers that, when multiplied, produce a.
  2. Write out all the pairs of numbers that, when multiplied, produce c.
  3. Pick one of the a pairs -- (a1, a2) -- and one of the c pairs -- (c1, c2).
  4. If c > 0: Compute a1c1 + a2c2. If | a1c1 + a2c2| = b, then the factored form of the quadratic is
    1. (a1x + c2)(a2x + c1) if b > 0.
    2. (a1x - c2)(a2x - c1) if b < 0.
  5. If a1c1 + a2c2≠b, compute a1c2 + a2c1. If a1c2 + a2c1 = b, then the factored form of the quadratic is (a1x + c1)(a2x + c2) or (a1x + c1)(a2x + c2). If a1c2 + a2c1≠b, pick another set of pairs.
  6. If c < 0: Compute a1c1 -a2c2. If | a1c1 - a2c2| = b, then the factored form of the quadratic is:

    (a1x - c2)(a2x + c1) where a1c1 > a2c2 if b > 0 and a1c1 < a2c2 if b < 0.
Using FOIL, the outside pair plus (or minus) the inside pair must equal b.

  1. Check.


Example 1: Factor 3x2 - 8x + 4.

  1. Numbers that produce 3: (1, 3).
  2. Numbers that produce 4: (1, 4), (2, 2).
    • (1, 3) and (1, 4): 1(1) + 3(4) = 11≠8. 1(4) + 3(1) = 7≠ = 8.
    • (1, 3) and (2, 2): 1(2) + 3(2) = 8.
    • (x - 2)(3x - 2).
  4. Check: (x - 2)(3x - 2) = 3x2 -2x - 6x + 4 = 3x2 - 8x + 4.


Example 2: Factor 12x2 + 17x + 6.

  1. Numbers that produce 12: (1, 12), (2, 6), (3, 4).
  2. Numbers that produce 6: (1, 6), (2, 3).
    • (1, 12) and (1, 6): 1(1) + 12(6) = 72. 1(6) + 12(1) = 18.
    • (1, 12) and (2, 3): 1(2) + 12(3) = 38. 1(3) + 12(2) = 27.
    • (2, 6) and (1, 6): 2(1) + 6(6) = 38. 2(6) + 6(1) = 18.
    • (2, 6) and (2, 3): 2(2) + 6(3) = 22. 2(3) + 6(2) = 18.
    • (3, 4) and (1, 6): 3(1) + 4(6) = 27. 3(6) + 4(1) = 22.
    • (3, 4) and (2, 3): 3(2) + 4(3) = 18. 3(3) + 4(2) = 17.
    (3x + 2)(4x + 3).
  4. Check: (3x + 2)(4x + 3) = 12x2 +9x + 8x + 6 = 12x2 + 17x + 6.


Example 3: Factor 4x2 - 5x - 21.

  1. Numbers that produce 4: (1, 4), (2, 2).
  2. Numbers that produce 21: (1, 21), (3, 7).
    • (1, 4) and (1, 21): 1(1) -4(21) = - 83. 1(21) - 4(1) = 17.
    • (1, 4) and (3, 7): 1(3) - 4(7) = - 25. 1(7) - 4(3) = - 5.
    (x - 3)(4x + 7).
  4. Check: (x - 3)(4x + 7) = 4x2 +7x - 12x - 21 = 4x2 - 5x - 21.

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