The Discriminant
Simple Explanation
The discriminant, D = b² − 4ac, is a single number computed from a quadratic's coefficients that reveals how many times its graph crosses the x-axis: D > 0 means two real roots (two crossings), D = 0 means exactly one real root (the vertex touches the axis), and D < 0 means no real roots (the graph never crosses the axis).
Why Do We Need It?
The discriminant lets you know how many solutions a quadratic equation has before doing any further work to find them — a fast diagnostic check.
See It
An upward-opening parabola crossing the x-axis at two points, x = 2 and x = 3, illustrating a positive discriminant
Formula
Discriminant of a Quadratic
D = b² − 4ac
A single number, computed from a quadratic's coefficients, that reveals how many real roots (x-intercepts) the quadratic has — without solving the equation.
- D
- — the discriminant
- a, b, c
- — the coefficients of ax² + bx + c = 0 (a ≠ 0)
When to use it: Whenever you need to know how many real solutions a quadratic equation has (two, one, or none) before — or instead of — actually solving it.
Worked Example
Use the discriminant to count roots
How many real roots does 2x² + 3x + 5 = 0 have?
Why Does This Work?
The discriminant is exactly the expression under the square root in the quadratic formula — a negative value under a square root has no real result (no real roots), zero gives one repeated result (one root), and a positive value gives two distinct results (two roots).
Real-Life Example
Checking whether a projectile ever reaches a target height
A projectile's height is modeled by a quadratic, and you want to know whether it ever reaches a specific target height without solving the full equation.
Setting the height equation equal to the target and checking the discriminant instantly tells you whether the projectile reaches that height (D ≥ 0) or never does (D < 0).
Practice
How many real roots does x² − 6x + 9 = 0 have?
MediumCommon mistake
Forgetting the negative sign in −4ac, or mixing up b² with (bx)² — the discriminant uses only the coefficient b, squared, not the whole term bx.
Quick Review
- D = b² − 4ac.
- D > 0: two real roots. D = 0: one repeated real root. D < 0: no real roots.
- The discriminant tells you how many times the graph crosses the x-axis, without solving.