How do you solve quadratic inequalities?


Quick Answer

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.

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Full Answer

  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.

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