Quadratic Formula

The General Quadratic Equation

This equation

is the general form of what’s called a second-degree or quadratic equation. A quadratic equation is any equation in which the highest power or exponent of the variable x is 2.

Examples:

In the first equation, a = 1, b = 2, and c = 1. In the second equation, a = 3, b = 4, and c = -3. In the third equation, a = 5, b = -25, and c = -14.

A quadratic equation doesn’t have to have the general form, but it can be put into that form. For example, suppose you have this equation:

Then by adding 2 to each side of the equation you get:

and then multiplying each side of the equation by -1, you get:

which is in the general form, with a = 5, b = -3, and c = -2.

Quadratic Formula

Every quadratic equation can be solved by simply putting the equation into the general form

and then plugging the values for a, b, and c into this formula:

The

sign means, “plus or minus.” This means that there are actually two solutions defined here – one where a plus sign is placed before the result of the radical, and one where a minus sign is placed before the result. Here are some examples:

Using the quadratic formula, with a = 1, b = 3, and c = 1, we get

so the solutions are

which equals

and

Second example:

In this case, a = 1, b = 4, and c = 5, so

and since

then the solutions from the quadratic formula are:

Third example:

In this example,

so therefore the solutions are

which computes to x = -1 and x = -3.