A quadratic equation has the form ax² + bx + c = 0, and it describes any situation where a quantity depends on the square of another — the path of a thrown ball, the area of a rectangle with a fixed perimeter, the point where revenue peaks. Solving it means finding the values of x where the curve crosses zero.
The quadratic formula solves every one of them: x = [−b ± √(b² − 4ac)] ÷ 2a. The ± is what produces two solutions, corresponding to the two places a parabola can cross the horizontal axis.
The discriminant tells you what to expect
| b² − 4ac | Real solutions | What the graph does |
|---|---|---|
| Positive | Two distinct | Crosses the x-axis twice |
| Zero | One repeated | Touches the axis at its vertex |
| Negative | None (two complex) | Never reaches the axis |
Choosing a method
- Factoring: fastest when the roots are whole numbers. x² + 5x + 6 factors to (x+2)(x+3), giving −2 and −3.
- Quadratic formula: always works, no inspection required, and the sensible default.
- Completing the square: slower to solve with, but it is how the formula is derived and it converts the equation into vertex form.
- Graphing: shows the roots as intercepts and is the best way to build intuition, though imprecise for exact values.
Reading the parabola
The sign of a decides the direction: positive opens upward with a minimum, negative opens downward with a maximum. The vertex sits at x = −b ÷ 2a, and substituting that back gives the extreme value — which is what you want in any optimisation problem. The curve is symmetric about the vertical line through the vertex, so the two roots are always equidistant from it.
Where quadratics show up
- Projectile motion: height against time under constant gravity is quadratic, so the formula gives the landing time.
- Optimisation: maximum profit or minimum cost when the relationship is quadratic sits at the vertex.
- Geometry: area problems with a constrained perimeter reduce to a quadratic.
- Braking distance: it scales with the square of speed, which is why a modest speed increase lengthens stopping distance so much.
Related tools
- Square Root Calculator — the discriminant needs one
- Scientific Calculator
- Fraction Calculator
- Browse all free tools