How To Solve Quadratic Equations

When working with quadratic equations it is often very useful to find the roots of the equations (these are the values of x that make the equation equal to 0). We refer to finding the roots as solving the equation (solving for the roots). There are 4 different ways to solve quadratic equations. Each of them has there own merits and uses.

FACTOR:

Many quadratic equations can be factored into two different parts. For example,

x +3x – 4 = 0  can be represented as
(x – 4)(x + 1) = 0

We could convert back to our full quadratic by FOILing.

Now, since we are multiplying these two terms, and there product is 0, one of these two terms must be equal to 0. So

x – 4 = 0  or
x + 1 = 0

These become
x = 4 and x = -1.

These are the solutions to the original quadratic equation. Factoring is one of the quickest methods and very useful when working with quadratic equations that have small integers as roots.

QUADRATIC FORMULA:

All quadratic equations can be represented in the following form:

ax + bx + c = 0, where a, b, and c are all integers.

If the roots of a quadratic equation are not integers (this is very often the case when the coefficient "a" is not 1), factoring can be nearly impossible. The Quadratic Formula is extremely useful in this case. The Quadratic Formula states that the two roots of a quadratic equation are

roots = (1/2a)[-b ± (b – 4ac)]

Notice the ± sign. These means plus or minus. One root will be found by finding the sum and the other by finding the difference. The Quadratic Formula is a little time consuming but it always works when the roots exist. It is worth memorizing.

GRAPHING

If you have access to a graphing calculator then graphing is an extremely useful method for finding roots. Simply graph the function and then find the values where the parabola crosses the x-axis. These are the roots! Again, extremely fast and always finds the roots, but not practical since you will often be forced to find roots by non-calculator methods.

COMPLETING THE SQUARE

Completing the Square is a final method for finding the roots of a quadratic equation. It is very time consuming and is actually much more useful for finding the vertex. There is a separate tutorial for Completing the Square.