How do you solve quadratic inequalities?

To solve a quadratic inequality, find the coordinates where the graph crosses the x axis, and determine the direction for the parabola. Complete the parabola through the points and shade the appropriate points to express the solution graphically.

1. Find the "zeros" of the equation

   Remember that quadratic equations and inequalities appear on graph paper as a parabola, so find the two points where the parabola for the inequality crosses the x-axis, or the "zeroes" (points at which x = 0). Consider the example problem: Solve -x^2 + 9 > 0. Turn the inequality into an equation: -x^2 + 9 = 0. Multiply both sides of the equation by (-1) to yield x^2 - 9 = 0. Factor the quadratic using the difference of two squares method (x^2 is the square of x, and 9 is the square of 3) to yield (x+3)(x-3) = 0. Write the solution: x = 3 or x = -3. Draw points on a graph at (3,0) and (-3,0).

2. Determine the direction of the parabola

   Look at the original inequality to determine whether the parabola points upward or downward. Consider the example problem again: -x^2 + 9 > 0. Plan for a parabola that points downward because the constant multiplied by the quadratic term (the x^2) is negative (-1).

3. Finish solving the inequality

   Draw the parabola, running a curve that is above the x-axis between the zeros (3 and -3) and then slopes downward on both sides of the y-axis, running through (0,3) and (0-3) as well. Find the intervals where the graph is above the x-axis, because the y values must be greater than zero according to the original inequality. Shade in the area beneath the parabola but above the x-axis, as this is the portion of the parabola that keeps y over zero and has x between -3 and 3.