Solving by Factoring

Solving a quadratic (or any kind of equation) by factoring it makes use of a principle known as the zero-product rule.

 

Zero Product Rule If ab = 0 then either a = 0 or b = 0 (or both). In other words, if the product of two things is zero then one of those two things must be zero, because the only way to multiply something and get zero is to multiply it by zero. Thus, if you can factor an expression that is equal to zero, then you can set each factor equal to zero and solve it for the unknown. ·         The expression must be set equal to zero to use this principle ·         You can always make any equation equal to zero by moving all the terms to one side. Example: Given: x2 – x = 6 Move all terms to one side: x2 – x – 6 = 0 Factor: (x – 3)(x + 2) = 0 Set each factor equal to zero and solve: (x – 3) = 0   OR   (x + 2) = 0 Solutions: x = 3 OR   x = -2

 

No Constant Term

If a quadratic equation has no constant term (i.e. c = 0) then it can easily be solved by factoring out the common x from the remaining two terms:

Then, using the zero-product rule, you set each factor equal to zero and solve to get the two solutions:

x = 0   or   ax + b = 0

x = 0   or   x = –b/a

 

WARNING: Do not divide out the common factor of x or you will lose the x = 0 solution. Keep all the factors and use the zero-product rule to get the solutions.

Trinomials

When a quadratic has all three terms, you can still solve it with the zero-product rule if you are able to factor the trinomial.

·                     Remember, not all trinomial quadratics can be factored with integer constants

If it can be factored, then it can be written as a product of two binomials. The zero-product rule can then be used to set each of these factors equal to zero, resulting in two equations that are both simple linear equations that can be solved for x. See the above example for the zero-product rule to see how this works.

A more thorough discussion of factoring trinomials may be found in the chapter on polynomials, but here is a quick review:

Tips for Factoring Trinomials

1.                  Clear fractions (by multiplying through by the common denominator)

2.                  Remove common factors if possible

3.                  If the coefficient of the x² term is 1, then

x² + bx + c = (x + n)(x + m), where n and m

                                             i.      Multiply to give c

                                           ii.      Add to give b

4.                  If the coefficient of the x² term is not 1, then use either

a.       Guess-and Check

                                                   i.      List the factors of the coefficient of the x² term

                                                 ii.      List the factors of the constant term

                                                iii.      Test all the possible binomials you can make from these factors

b.      Factoring by Grouping

                                                   i.      Find the product ac

                                                 ii.      Find two factors of ac that add to give b

                                                iii.      Split the middle term into the sum of two terms, using these two factors

                                               iv.      Group the terms into pairs

                                                v.      Factor out the common binomial