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EnchantedLearning.com Algebra: Quadratic Equations

A quadratic equation can be written in the form:

,
where a, b, and c are numbers (a ≠0), and x is the variable. x is a solution (or a root) if it satisfies the equation ax² + bx + c = 0.

Some examples of quadratic equations include:

3x2 + 9x - 2 = 0

6x2 + 11x = 7

4x2 = 13

x2 - 1 = 0 (x + 1)(x - 1) = 0 x = ±1

x2 - 5x + 6 = 0 (x - 2)(x - 3) = 0 x = 2, 3

Most second-degree equations are more difficult to solve, and cannot be solved by simple factoring. The quadratic formula is a general way of solving any quadratic equation:

This formula gives two solutions, although the two solutions may be the same number. (When solving any polynomial equation of degree n, there are at most n solutions to that equation.)

Divide each side of the equation by a.

Subtract c/a from each side of the equation.

Add (b/2a)² to each side of the equation (to complete the square).

Find a common denominator for the right side of the equation.

Take the square root of each side of the equation.


Add b/2a to each side of the equation.


The plus-or-minus sign shows that there are two possible solutions.

The green parabola has 2 x-intercepts. Its corresponding quadratic equation has 2 distinct solutions (x=1 and x=4). The yellow parabola has 1 x-intercept. Its corresponding quadratic equation has 1 solution (x=-3). The purple parabola has no x-intercepts. Its corresponding quadratic equation has no solutions.

Every quadratic equation has at most two solutions, but for some equations, the two solutions are the same number, and for others, there is no solution on the number line (because it would involve the square root of a negative number).

One way to understand this visually is to realize that when you graph a quadratic equation (a second-degree equation), you get a parabola. When you set the quadratic equation equal to zero, this represents the points where the parabola hits the x-axis (the x-intercepts, where y=0).

Quadratic Equations Worksheets (with no linear term) Solve simple (pure) quadratic equations (no linear terms).

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