The discriminant may be used to solve quadratic inequalities of the form $ ax^2+bx+c > 0 (\text{or} \ge 0, \lt 0, \le 0)$, using one of the following rules:
1) If the discriminant $b^2-4c$ is negative, the quadratic expression $ax^2+bx+c$ takes the sign of coefficient $a$ for $x$ in the interval $(-\infty, +\infty)$. 2) If the discriminant $b^2-4c$ is equal to zero, the quadratic expression $ax^2+bx+c$ takes the sign of coefficient $a$ for $x$ in the intervals $(-\infty, -b/2a)$ and $(-b/2a,+\infty)$. The expression $ax^2+bx+c$ is equal to zero at $x = -b/2a$. 3) If the discriminant $b^2-4c$ is positive, the quadratic expression $ax^2+bx+c$ has two zeros $x_1$ and $x_2$ (assuming $x_1 < x_2$). $ax^2+bx+c$ has the same sign as coeffificent $a$ for $x$ in the intervals $(-\infty, x_1)$ and $(x_2 , \infty)$ and opposite sign to that of coefficient $a$ in the interval $(x_1 , x_2)$. The expression $ax^2+bx+c$ is equal to zero at $x = x_1$ and $x = x_2$. Solve each step below then click on "Show me" to check your answer. There is a graph at the bottom of the page that helps you further understand graphically the solution to the question shown below. Step by step solutionBelow is shown the graph of the quadratic expression on the right hand side of the inequality in step 2. Check that the analytical and graphical solution shown below agree. (Change scales if necessary) |