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 ${\displaystyle x^{2}}$ is positive, then find the x-intercept, and use knowledge of whether the parabola opens up or down to solve the problem.
Solve the inequality ${\displaystyle 3x^{2}-5x+2\geq 0}$
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 ${\displaystyle (-5)^{2}-4(3)(2)=1}$. 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 ${\displaystyle {\frac {5\pm {\sqrt {1}}}{2(3)}}={\frac {5\pm 1}{6}}}$. So the two x-intercepts are ${\displaystyle (1,0),{\text{ and }}({\frac {2}{3}},0)}$.
Since the coefficient of ${\displaystyle x^{2}}$ is positive the parabola opens up. Thus, the answer is ${\displaystyle (-\infty ,{\frac {2}{3}}]\cup [1,\infty )}$