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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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.
Example
Solve the inequality Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 3x^2 - 5x + 2 \ge 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{5 \pm\sqrt{1}}{2(3)} = \frac{5\pm 1}{6}} . So the two x-intercepts are Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1, 0), and (\frac{2}{3}, 0)} .
Since the coefficient of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^2} is positive the parabola opens up. Thus, the answer is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, \frac{2}{3}]\cup [1, \infty)}