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Algebra II: Factoring

Factoring ax 2 + bx + c

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Factoring ax 2 + bx + c

This section explains how to factor expressions of the form ax 2 + 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 (ax 2 + 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 ax 2 + 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 -- (a 1, a 2) -- and one of the c pairs -- (c 1, c 2) .
  4. If c > 0 : Compute a 1 c 1 + a 2 c 2 . If | a 1 c 1 + a 2 c 2| = b , then the factored form of the quadratic is
    1. (a 1 x + c 2)(a 2 x + c 1) if b > 0 .
    2. (a 1 x - c 2)(a 2 x - c 1) if b < 0 .
  5. If a 1 c 1 + a 2 c 2b , compute a 1 c 2 + a 2 c 1 . If a 1 c 2 + a 2 c 1 = b , then the factored form of the quadratic is (a 1 x + c 1)(a 2 x + c 2) or (a 1 x + c 1)(a 2 x + c 2) . If a 1 c 2 + a 2 c 1b , pick another set of pairs.
  6. If c < 0 : Compute a 1 c 1 -a 2 c 2 . If | a 1 c 1 - a 2 c 2| = b , then the factored form of the quadratic is:

    (a 1 x - c 2)(a 2 x + c 1) where a 1 c 1 > a 2 c 2 if b > 0 and a 1 c 1 < a 2 c 2 if b < 0 .
Using FOIL, the outside pair plus (or minus) the inside pair must equal b .

  1. Check.


Example 1: Factor 3x 2 - 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) .
  3. Check: (x - 2)(3x - 2) = 3x 2 -2x - 6x + 4 = 3x 2 - 8x + 4 .


Example 2: Factor 12x 2 + 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) .
  3. Check: (3x + 2)(4x + 3) = 12x 2 +9x + 8x + 6 = 12x 2 + 17x + 6 .


Example 3: Factor 4x 2 - 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) .
  3. Check: (x - 3)(4x + 7) = 4x 2 +7x - 12x - 21 = 4x 2 - 5x - 21 .

Any easy way to check your answer is to pick a value of x ( x = 10 is a good value to pick) and evaluate the original expression and the factored form. You should get the same result for both expressions.

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