3.4 Solving Quadratic Inequalities h286o

Solving Quadratic Inequalities

Graphical Solution of Quadratic Inequalities You already know that the graph of the equation y=ax2+bx+c is a parabola. You also know that the x-intercepts of this parabola are exactly the roots of the associated quadratic equation ax2+bx+c = 0. These are the x-coordinates of the point(s) where the parabola y=ax2+bx+c intersects the line y=0, that is, the x-axis.

Suppose you want to solve the quadratic inequality ax2+bx+c ˂ 0 Then this is the same as asking yourself: For what values of x will the parabola y=ax2+bx+c lie below the line y=0 (the x-axis)?

Example Suppose you want to solve this quadratic inequality. x2-2x-3 ˂ 0 If you sketch the graph of the associated parabola y = x2-2x-3, you will get something like this:

Where does the parabola intersect the xaxis? For what values of x does thee parabola lie below the x-axis? It is clear that the graph is below the x=axis on the interval between the two roots of the quadratic equation. Hence, the inequality x2-2x-3 ˂ 0 is satisfied when x is in the interval (-1, 3). Next consider the inequality x2-2x-3 ˃ 0. The solution consists of all values of x for which the parabola y = x2-2x-3 is either on the x-axis or above it, since the inequality involves a greater than or equal to relation. Examine the parabola above and determine these values. You will see that the parabola is above the x-axis to the left of x=-1 and to the right of x=3, and intersect the x-axis at x=-1 and x=3. This shows that the solution to this inequality consists of all values satisfying x˂-1

Our example illustrate the following procedure for solving quadratic inequalities: to obtain the graphical solution to a quadratic inequality, do the following: 1. Write the inequality in one of the following forms: ax2+bx+c ˃ 0 ax2+bx+c ˂ 0 ax2+bx+c ˃ 0 ax2+bx+c ˂ 0 2. Find the roots of the associated quadratic equation ax2+bx+c = 0 say x1 and x2, where x1˂x2. 3. Sketch the graph of the parabola y = ax2+bx+c. Use the graph to determine on which of the following intervals the graph will be above or below the x-axis: x˂x1,

Procedure for Solving Quadratic Inequalities To solve a quadratic, do the following: 1. Rewrite the inequality such that the lefthand side takes the form ax2+bx+c and the right-hand side is zero. 2. Solve for the roots of the associated quadratic equation ax2+bx+c=0. Denote the roots by x1 and x2. 3. Rewrite the left-hand side of the inequality in the form a(x-x1)(x-x2). 4. Subdivide the real line into subintervals using the roots obtained in (b) as division points. In each interval, determine the sign of a(x-x1)(x-x2) by considering the signs of the factors. 5. The solution set consists of the intervals that give the appropriate sign for a(x-x1)(x-