Quadratic Inequalities

Introduction

In this section we will solve inequalities that involve quadratic functions. The approach can be summarized as moving everything onto one side of the inequality sign, preferably so the coefficient of $x^{2}$ is positive, then find the x-intercept, and use knowledge of whether the parabola opens up or down to solve the problem.

Example

Solve the inequality $3x^{2}-5x+2\geq 0$

Since all of the non-zero terms are already on the same side we can skip the first step. Now we need to look at the discriminant. The discriminant is $(-5)^{2}-4(3)(2)=1$. So there are two distinct x-intercepts(zeroes) and we can use the quadratic formula to find them. By the quadratic formula the x-intercepts are ${\frac {5\pm {\sqrt {1}}}{2(3)}}={\frac {5\pm 1}{6}}$. So the two x-intercepts are $(1,0),{\text{ and }}({\frac {2}{3}},0)$.

Since the coefficient of $x^{2}$ is positive the parabola opens up. Thus, the answer is $(-\infty ,{\frac {2}{3}}]\cup [1,\infty )$

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