Examples of two (of many) types of quadratic equations are shown below:

• y = ax² + bx + c.
• ax² + bx + c = 0.

The first equation above is a quadratic equation in two variables, while the second equation above is a quadratic equation in one variable.

Important Distinction

Although a quadratic function is defined as any function which can be put into the form f(x) = ax² + bx + c, people generally also refer to y = ax² + bx + c as a quadratic function, even when function notation isn't used.

For consistency within this glossary, f(x) = ax² + bx + c and y = ax² + bx + c will be referred to as quadratic functions, while ax² + bx + c = 0 will be referred to as a quadratic equation.

When a quadratic function is graphed in a coordinate plane, the resulting curve is a parabola.

When a quadratic equation is graphed in a coordinate plane, the resulting graph is two, one or zero vertical lines. The number of vertical lines, as well as the position of the lines, are related to the x-intercept(s) of the quadratic function graphed in the coordinate plane.

Because a quadratic equation has a single variable, it is usually NOT graphed in the coordinate plane. Usually the solution set of the quadratic equation is stated, and if the solution set is graphed, it is graphed along a number line.

When "multiples" of a quadratic function are graphed, it can be seen that a family of parabolas occur that have equivalent x-intercepts.

For example, observe the graphs of the following quadratic functions, all graphed in the same coordinate plane.

• y = x² - 4x + 3
• y = 3x² - 12x + 9        or        y = 3[ x² - 4x + 3 ]
• y = -2x² + 8x - 6        or        y = -2[ x² - 4x + 3 ]
• y = -4x² + 16x - 12        or        y = -4[ x² - 4x + 3 ]

Note that when the variable y is set to zero, all of the quadratic functions become quadratic equations (that all reduce to the same quadratic equation).

The solution set to the given quadratic equation is { 1, 3 }, the common x-intercepts of all of the quadratic functions.

The main algebraic ways of solving a quadratic equation are:

• by determining simple square roots (this method works best when b = 0)
• by factoring (this method works best when the discriminant equals zero (Δ = 0)
• by completing the square
• by applying the quadratic formula.

It is possible to estimate the solutions to the quadratic equation by graphing a corresponding quadratic function and looking at the x-intercepts.