Factoring Quadratics

A "quadratic" is a polynomial that looks like "ax2+ bx + c", where "a", "b", and "c" are just numbers.

In the case of factoring problems, you will find two numbers that will not only multiply to equal the constant term "c", but also add up to equal "b", the coefficient on the x-term. For instance:

• Factor x2 + 5x + 6.

            The product of 2 and 3 equals 6 matching the constant term, and the sum of those two numbers equals the "b" as the coefficient of the x-term. 

            So the factors are: (x+2) and (x+3)

• Factor x2 + 7x + 6.

  • The constant term is 6, which can be written as the product of 2 and 3 or of 1 and 6.  But 2 + 3 = 5, so 2 and 3 are not the numbers you need in this case. On the other hand, 1 + 6 = 7, those would be the factors: (x+6) and (x+1)

• Note that the order doesn't matter in multiplication, so the above answer could also be correctly written as "(x + 6)(x + 1)"

• Factor x2 – 5x + 6.

  • The constant term is 6, but the middle coefficient this time is negative. Since you multiplied to a positive six, then the factors must have the same sign. Since you're adding to a negative (–5), then both factors must be negative. So rather than using 2 and 3, as in the first example, this time you will use –2 and –3:

• x2 – 5x + 6 = (x – 2)(x – 3)

Note that you can use clues from the signs to determine which factors to use, as in this last example above:

• • If c is positive, then the factors you're looking for are either both positive or else both negative.

  • If b is positive, then the factors are positive

  • If b is negative, then the factors are negative.

  • In either case, you're looking for factors that add to b.

• If c is negative, then the factors you're looking for are alternating signs;

• that is, one is negative and one is positive.

• If b is positive, then the larger factor is positive.

• If b is negative, then the larger factor is negative.

• In either case, you're looking for factors that are b units apart.