Solving Quadratic Inequalities
Solve inequalities such as x² - x - 6 > 0 by finding the roots, sketching the parabola, and reading off the x values where the curve is on the required side of the axis.
What a learner can do afterwards
- Find the roots first, then sketch to decide which regions satisfy the inequality
- Write the answer as one or two ranges with the correct strict or inclusive signs
- Explain why an upward parabola gives two separate ranges for > 0
1 · Read
You have already solved linear inequalities: same balancing moves as an equation, flip the sign when you divide by a negative, and the answer is a stretch of the number line with an open or closed endpoint. A quadratic inequality looks the same on the outside, but the balancing moves run out of steam. You cannot just divide by x squared, because x squared can be zero. So the method changes shape: find the roots of the matching equation, sketch the parabola, and read off which stretches of the x axis put the curve on the right side of the axis.
The method has four moves, and the roots do all the heavy lifting. First, replace the inequality sign with an equals sign and solve that quadratic equation, by whatever method fits. Those roots are the only x values where the curve can cross the axis, so they are the boundaries of your answer. Second, decide the shape: if the x squared term is positive the parabola opens up, and if it is negative it opens down. If the x squared term is negative, multiply the whole inequality by -1 and flip the sign first, so the curve opens up and the shape rules below just work.
Third, check the middle: pick a test value between the two roots, often zero, and see which side of the axis it lands on. Fourth, read off the ranges. For a U shape that is above the axis, the answer is the two stretches outside the roots. For a U shape below the axis, it is the single stretch between them. An n shape flips both of those. The sign of the inequality decides which side you want, and the shape of the curve decides where that side is.
Solve x squared - 3x - 4 > 0. Set the equals version: x squared - 3x - 4 = 0. The pair that multiplies to -4 and adds to -3 is -4 and 1, so the factors are (x - 4)(x + 1) and the roots are 4 and -1. The curve opens up. Test the middle: x = 0 gives -4, below zero, so the curve sits under the axis between the roots. The inequality wants above the axis, so the answer is the two outer stretches: x < -1 or x > 4. Both signs are strict, so both endpoints stay open.
Open circles at -1 and 4, both outer stretches shaded.
Now the other side of the axis. Solve x squared + x - 6 < 0. The equals version factors as (x + 3)(x - 2), so the roots are -3 and 2. The curve opens up, and you want the part that is below the axis, which for a U shape is the middle stretch. The answer is -3 < x < 2. Notice the difference in shape: a below-the-axis U shape gives one range between the roots, and an above-the-axis U shape gives two ranges outside them.
Open circles at -3 and 2, the stretch between them shaded.
The roots can collapse. Solve x squared + 8x + 16 < 0. The equals version is (x + 4) squared = 0, a repeated root at -4. The curve touches the axis there and never dips below it, so there is no x that makes it strictly less than zero. No solution. If the question had asked x squared + 8x + 16 >= 0 instead, the answer would be every x, because the curve is above the axis everywhere except the touching point. And if an inequality has no roots at all, the curve never crosses the axis, so the whole line is either in or out.
Solve the matching equation to find the roots, sketch the parabola in your head, and read off the ranges where the curve is on the required side of the axis, keeping endpoints open or closed to match the sign.
More to read
Key Concepts: Quadratic Functions
A quadratic function is written like y = ax^2 + bx + c. Its graph is a parabola, a U-shaped curve that opens up or down depending on the sign of a. The vertex is the highest or lowest point on the curve, and the axis of symmetry is the vertical line that cuts the parabola into two mirror-image halves. The x-intercepts, also called zeros or roots, are the points where the parabola crosses the x-axis. Finding these roots is your first step whenever you sketch a parabola or solve a quadratic inequality, because they show exactly where the curve switches from positive to negative.
Understanding the Quadratic Formula
The quadratic formula helps you find the roots of any equation shaped like ax^2 + bx + c = 0. The formula is x equals negative b, plus or minus the square root of b squared minus 4ac, all over 2a. Plug in your values of a, b, and c, and it hands you the x-values where the parabola crosses the x-axis. Those roots are exactly what you need before you can sketch the parabola and figure out which regions satisfy a quadratic inequality like x^2 - x - 6 > 0.
Using Symmetry to Graph Quadratic Functions
A quadratic function in standard form looks like f(x) = ax^2 + bx + c. The value of a tells you which way the parabola opens. If a is greater than 0, it opens upward, like a smile. If a is less than 0, it opens downward, like a frown. The value of b shifts where the vertex sits along the x-axis, and c is the y-intercept, the value of the function when x is 0. Every parabola is symmetric about a vertical line called the axis of symmetry, which runs straight through the vertex. Knowing whether your parabola opens up or down is exactly what explains why an upward-opening one gives you two separate ranges when you solve something like x^2 - x - 6 > 0.
2 · Watch
به انگلیسی3 · Play
Pick the side you want, above or below the axis, and read off whether the answer is one range or two.
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Where it sits
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
34 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.