Solving the quadratic equation x²+bx+c=0 gives you two roots. But what if there's an inequality sign (>, <, ≥, ≤) instead of an equals sign? An expression like that is called a quadratic inequality. It might look harder at first glance, but if you already know how to solve a quadratic equation, picturing just one graph is all it takes to solve it quickly.
The most important tool for solving a quadratic inequality is its graph. Picture the quadratic function y=x²+bx+c. Its graph, remember, is a smooth curve shaped like a U or an upside-down U — a parabola. The solution to the quadratic inequality x²+bx+c > 0 is really the same question as "for which x values does this parabola sit above the x-axis?" Conversely, for < 0, you're looking for the x values where the parabola sits below the x-axis.
So how can you tell where the parabola is above the x-axis and where it's below? The two roots of the quadratic equation are exactly the hint you need. The two roots are the points where the parabola meets the x-axis precisely — in other words, the boundaries where the sign changes. For example, factoring x²−x−6=0 gives (x+2)(x−3)=0, so the two roots are x=−2 and x=3. Using these two points as boundaries splits the graph into three intervals: x<−2, −2<x<3, and x>3.
Whether the graph opens upward or downward plays the decisive role here. If a>0, so it opens upward (∪ shape), the parabola dips down (negative) between the two roots, then rises back up (positive) outside them. That's why the solution to x²−x−6>0 is the region outside the two roots, x<−2 or x>3, while the solution to x²−x−6<0 is the region between the two roots, −2<x<3.
If instead a<0, so the graph opens downward (∩ shape), it's exactly the opposite: the graph sits above the x-axis between the two roots, and below outside them. This is exactly where it's easy to make a mistake when solving a quadratic inequality — if you memorize "outside is the solution" without checking the sign of coefficient a, you'll get it wrong whenever a is negative.
You also can't overlook whether the inequality includes an equals sign (≥, ≤). With a strict inequality (>, <), the two boundary roots themselves are not part of the solution, because the expression equals exactly 0 at those points. But with ≥ or ≤, the two roots do need to be included in the solution.
There's also the case where the two roots coincide into one (a repeated root). Here, the parabola just barely touches the x-axis and then rises or falls again, so apart from that single point of contact, the sign is the same for every other x. For example, the solution to (x−1)² > 0 is "all real numbers except x=1," and there's no x at all that satisfies (x−1)² < 0 — squaring any number can never give a negative result.
To sum up, solving a quadratic inequality comes down to: ① find the two roots by solving the quadratic equation, ② check whether the graph opens upward or downward, then ③ pick the interval that matches the direction of the inequality sign. Rather than forcing yourself to memorize a rule, getting into the habit of sketching the parabola makes it much less confusing.
On our activity page, you can use sliders to change the two roots and the direction the parabola opens, and watch how the solution interval changes right on the graph — and you can also practice solving quadratic inequalities yourself with the quiz.