SOLUTION BY FACTORING
The equation x2 - 36 = 0 is a pure quadraticequation, There are two numbers which, when substituted for x, will satisfy the equation as follows:
Thus, +6 and - 6 are roots of the equation
x2 - 36 = 0
The most direct way to solve a pure quadratic (one in which no x term appears and the constant term is a perfect square) involves rewriting with the constant term in the right member, as follows:
X2 = 36
Taking square roots on both sides, we have
x = ±6
The reason for expressing the solution as bothplus and minus 6 is found in the fact that both +6 and -6, when squared, produce 36.
x2 - 36 = 0
can also be solved by factoring, as follows:
We now have the product of two factors equalto zero. According to the zero factor law, if a product is zero, then one or more of its factors is zero. Therefore, at least one of the factors must be zero, and it makes no difference which one. We are free to set first one factor and then the other factor equal to zero. In so doing we derive two solutions or roots of the equation. If x + 6 is the factor whose value is 0, then we have
If x - 6 is the zero factor, we have
When a three-term quadratic is put intosimplest form, it is customary to place all the terms on the left side of the equality sign with the squared term first, the-first-degree term next, and the constant term last, as in
9x2 - 2x+7=0
If the trinomial in the left member is readilyfactorable, the equation can be solved quickly by separating the trinomial into factors. Consider the equation
3x2 - x - 2 = 0
By factoring the trinomial, the equation becomes
(3x + 2)(x - 1) = 0
Once again we have two factors, the product ofwhich is 0. This means that one or the other of them (or both) must have the value 0. If the zero factor is 3x + 2, we have
If the zero factor is x - 1, we have
x - 1 = 0
Substituting first x = 1 and then x = -2/3 inthe original equation, we see that both roots satisfy it. Thus,
In summation, when a quadratic may bereadily factored, the process for finding its roots is as follows:.
1. Arrange the equation in the order of thedescending powers of the variable so that all the terms appear in the left member and zero appears in the right.
2. Factor the left member of the equation.
3. Set each factor containing the variableequal to zero and solve the resulting equations.
4. Check by substituting each of the derivedroots in the original equation.
EXAMPLE: Solve the equation x2 - 4x = 12 for x.
Practice problems. Solve the following equations by factoring: