# Free Mathematics Tutorials This is a tutorial on how to solve quadratic inequalities graphically. The quadratic inequalities explored are of the type

a x 2 + b x + c < 0
or
a x 2 + b x + c > 0

 Review An applet plots the graph of y = a x 2 + b x + c and displays part of the graph that is below the x axis (y < 0) in blue and part of the graph that is above the x axis (y > 0) in red. To solve a quadratic inequality you just read the interval corresponding to y < 0 or y > 0 depending on the inequality to solve. Interactive Tutorials click on the button above "click here to start" to start the applet and MAXIMIZE the window obtained. Your browser is completely ignoring the tag! Example 1 : Solve graphically and analytically the quadratic inequality - x 2 + 3x + 4 < 0 Solution to Example 1: Graphical solution: Use the applet to set coefficients a = -1, b = 3 and c = 4 and graph the equation y = - x 2 + 3x + 4. The solution set to the inequality - x 2 + 3x + 4 < 0 correspond to the x coordinates of the points on the graph for which y < 0 BLUE. We have two intervals for which y < 0: (-∞ , -1) U (4 , +∞) Analytical solution: Factor the left hand term of the given inequality - x 2 + 3x + 4 = (x + 1)(-x + 4) To solve the given inequality, we study the sign of (x + 1)(-x + 4) on the intervals (-∞ , -1), (-1 , 4) and (4 , +∞) where -1 and 4 are the zeros of (x + 1)(-x + 4). let x = -2 a value of x on the interval (-∞ , -1). For this value - x 2 + 3x + 4 = -(-2) 2 + 3(-2) + 4 = -6 is negative. The interval (-∞ , -1) is one solution set. Let x = 0 on the interval (-1 , 4). For this value - x 2 + 3x + 4 = 4 is positive. x = 5 is in the interval (4 , +∞). For this value of x, - x 2 + 3x + 4 = -6 is negative. The interval (4 , +∞) is a solution set for the given inequality. Hence the solution set of the inequality is given by the union of all intervals for which - x 2 + 3x + 4 is negative: (-∞ , -1)U (4 , +∞) Both the graphical and analytical methods give the same answer. Example 2 : Solve graphically and analytically the equation -x 2 + 4x - 5 > 0 Solution to Example 2: Graphical solution: Use the applet to set coefficients a = -1, b = 4 and c = -5 and graph the equation y = -x 2 + 4x - 5. This inequality has no solutions since the whole graph is below the x axis and therefore y < 0 everywhere. Analytical solution: -x 2 + 4x - 5 cannot be factored over the real numbers. Therefore -x 2 + 4x - 5 has no zeros and its sign does not change. To find the sign of -x 2 + 4x - 5 you need to evaluate it for one single value of x. Let us evaluate -x 2 + 4x - 5 at x = 0 -(0) 2 + 4(0) - 5 = - 5. -x 2 + 4x - 5 is always negative hence the inequality -x 2 + 4x - 5 > 0 has no solutions. Exercises: Solve graphically (using the applet) and analytically the following quadratic inequalities. 1: -x 2 - 4 x < -5 2: x 2 - 2 x + 8 >= 0 3: x 2 - 3 x <= 0 Solutions to Above Exercises: 1: (-∞ , -5) U (1 , +∞) 2: (-∞ , +∞) 3: [0 , 3] More references and links on how to Solve Equations, Systems of Equations and Inequalities.