How to solve quadratic inequalities by graphing
A quadratic inequality compares a quadratic expression to zero, such as ax² + bx + c > 0. The graphing method solves it by treating y = ax² + bx + c as a parabola: the inequality asks where that parabola lies above the x-axis (y > 0), below it (y < 0), or on it (y = 0). Because a parabola crosses the x-axis only at its roots, those roots split the number line into intervals where the sign of the expression never changes — so you only need to find the roots and check the direction the parabola opens to read off the full solution set.
Step-by-step method
- Find the roots. Solve ax² + bx + c = 0 with the quadratic formula, x = (−b ± √(b² − 4ac)) / 2a. The discriminant b² − 4ac tells you how many real roots exist: two if positive, one repeated root if zero, none if negative.
- Find the vertex. The axis of symmetry is x = −b / 2a; substitute it back into the expression for the vertex's y-value. This is the parabola's minimum (if a > 0) or maximum (if a < 0).
- Sketch the sign pattern. If a > 0 the parabola opens upward, so it is below the x-axis between the two roots and above the axis outside them. If a < 0 it opens downward, so those regions flip.
- Read off the interval. Match the sign pattern to the inequality: use open intervals (parentheses) for strict inequalities (>, <) since the roots themselves are excluded, and closed intervals (brackets) for non-strict inequalities (≥, ≤) since the roots are included.
Special cases to watch for
- No real roots (discriminant < 0): the parabola never touches the x-axis, so it is entirely one sign. The solution set is either all real numbers or the empty set, depending on the inequality direction and the sign of a.
- One repeated root (discriminant = 0): the parabola just touches the x-axis at a single point. That point satisfies ≥ or ≤ but never > or < on its own, and the rest of the number line is entirely one sign.
- Strict vs. non-strict: on a number-line sketch this is the difference between an open circle (root excluded, strict inequality) and a filled circle (root included, ≥ or ≤).
Frequently Asked Questions
How do you solve a quadratic inequality by graphing?
Rewrite the inequality as ax²+bx+c compared to 0, find the roots (x-intercepts) with the quadratic formula, and sketch the parabola. Since a parabola only crosses the x-axis at its roots, it is entirely above the axis on some intervals and entirely below it on others. Read off the intervals where the graph is above the axis (f(x) > 0) or below it (f(x) < 0) to get the solution set.
What does it mean if the discriminant is negative?
A negative discriminant (b²−4ac < 0) means the parabola never touches the x-axis, so it is entirely positive (if a > 0) or entirely negative (if a < 0) for every real x. In that case the inequality's solution set is either all real numbers or the empty set, depending on the inequality direction.
How do I write the solution in interval notation?
Order the roots x1 < x2, then use parentheses for strict inequalities (>, <) and brackets for non-strict inequalities (≥, ≤). For example, if a > 0 and the roots are 1 and 4, the solution to ax²+bx+c < 0 is the interval (1, 4), while ax²+bx+c ≤ 0 gives [1, 4].
Does it matter whether the parabola opens up or down?
Yes. When a > 0 the parabola opens upward, so it is below the x-axis between the roots and above the roots outside them. When a < 0 it opens downward, so those regions flip: the graph is above the axis between the roots and below it outside them.