Quadratic equations will have one term with a square (e.g.,
) and they take the form
+ Bx + C = 0
(A = 0),
A, B, and C are constants (e.g., numbers). A quadratic equation should always
be set to 0 before a solution is attempted. For instance, if we have an equation that is
+ 3x = -21, convert this equation to 2x
+ 3x + 21 = 0.
We will discuss two methods of solving a quadratic equation: by factoring and by the
Factoring to Solve a Quadratic Equation. Suppose we have the simple relation
= 0. We remember from above that this factors into (x - 1)(x + 1) = 0. This being
the case, at least one of the factors must equal 0. If this is not understood, realize that
there is no other way for the equation statement to be true. Keep in mind that anything
multiplied by 0 will be 0. So there are two solutions to the equation:
x - 1 = 0, thus x = 1 or x + 1 = 0 and x = -1.
Proof that these are correct answers is by substituting them in the equation.
x in this example:
- 100x + 2400 = 0.
This factors into
(x - 40)(x - 60) = 0. We now have two factors: x - 40 and x - 60,
whose product is 0. This means that we must have either
x - 40 = 0, where x = 40 or
x - 60 = 0, and in this case x = 60. We can check our results by substitution that either
of these values satisfies the equation.
Another example: Solve for
x. (x - 3)(x - 2) = 12. Multiply the factors: x
- 5x + 6,
- 5x + 6 = 12. Subtract 12 from both sides of the equation so that we set the
left hand side equal to 0. Thus:
- 5x - 6 = 0 factors into (x - 6)(x + 1) = 0.
x - 6 = 0, x = 6
x + 1 = 0, x = -1.
Quadratic Formula. This formula may be used on the conventional quadratic equation
in the generic form of
+ Bx + C = 0
(A = 0).
x is solved by simply manipulating the constants A, B, and C. The quadratic formula is
stated as follows:
x = [-B ± (B
or, rewritten with the radical sign:
Just like we did with the factoring method, the quadratic formula will produce two roots
(two answers): one with the plus before the radical sign and one with the minus before
the radical sign.