Factoring Quadratic Equations
Certain complete quadratic equations can be solved by factoring. If the left-hand side of the
general form of a quadratic equation can be factored, the only way for the factored equation to
be true is for one or both of the factors to be zero. For example, the left-hand side of the
quadratic equation x2 + x - 6 = 0 can be factored into (x + 3)(x - 2). The only way for the
equation (x + 3) (x - 2) = 0 to be true is for either (x + 3) or (x - 2) to be zero. Thus, the roots
of quadratic equations which can be factored can be found by setting each of the factors equal
to zero and solving the resulting linear equations. Thus, the roots of (x + 3)(x - 2) = 0 are found
by setting x + 3 and x - 2 equal to zero. The roots are x = -3 and x = 2.
Factoring estimates can be made on the basis that it is the reverse of multiplication. For
example, if we have two expressions (dx + c) and (cx + g) and multiply them, we obtain (using
the distribution laws)
(dx + c) (fx + g) = (dx) (fx) + (dx) (g) + (c) (fx) + cg =
= dfx2 + (dg + cf)x + cg.
Thus, a statement (dx + c) (fx + g) = 0 can be written
df x2 + (dg + cf)x + cg = 0.
Now, if one is given an equation ax2 + bx + c = 0, he knows that the symbol a is the product
of two numbers (df) and c is also the product of two numbers. For the example 3x2 - 4x - 4 =
0, it is a reasonable guess that the numbers multiplying x2 in the two factors are 3 and 1,
although they might be 1.5 and 2. The last -4 (c in the general equation) is the product of two
numbers (eg), perhaps -2 and 2 or -1 and 4. These combinations are tried to see which gives the
proper value of b (dg + ef), from above.
There are four steps used in solving quadratic equations by factoring.
Using the addition and subtraction axioms, arrange the equation in the
general quadratic form ax2 + bx + c = 0.
Factor the left-hand side of the equation.
Set each factor equal to zero and solve the resulting linear equations.
Check the roots.