Formula · Algebra
Discriminant Formula: What b² − 4ac Tells You
One expression, evaluated before you solve anything, that tells you what kind of answer you are about to get.
The formula, and where it sits
The discriminant formula is the part of the quadratic formula that lives under the square root: D = b² − 4ac, taken from a quadratic written in standard form as ax² + bx + c = 0.
It is worth evaluating on its own, first, because it decides the nature of the roots before any of the surrounding arithmetic is done. If it comes out negative you can stop — there is no real answer to find.
What each outcome means
There are three cases, and a fourth worth knowing that sits inside the first.
| b² − 4ac | Nature of the roots | The parabola |
|---|---|---|
| positive, a perfect square | two rational roots — it will factor | crosses the x-axis twice |
| positive, not a perfect square | two irrational roots | crosses the x-axis twice |
| zero | one repeated root | touches the x-axis once |
| negative | no real roots, a complex pair | never meets the x-axis |
The perfect-square case is the practically useful one. It tells you that factoring will work and will be faster than the formula, which turns the discriminant into a decision rather than just a description.
Worked examples of each case
- 2x² + 7x + 3 = 0a = 2, b = 7, c = 3
- 49 − 24 = 25a perfect square
- roots are −½ and −3rational, so it factors as (2x + 1)(x + 3)
- x² − x − 1 = 0a = 1, b = −1, c = −1
- 1 + 4 = 5positive, not a perfect square
- roots are (1 ± √5) ÷ 2irrational — the formula is required
- x² + 2x + 1 = 0a = 1, b = 2, c = 1
- 4 − 4 = 0zero discriminant
- x = −1 onlya perfect square trinomial, (x + 1)²
- x² + x + 1 = 0a = 1, b = 1, c = 1
- 1 − 4 = −3negative
- no real solutionthe roots are −½ ± (√3÷2)<i>i</i>
Reading it off the graph
The discriminant is a statement about where the parabola sits relative to the x-axis, and that makes it easy to check against a sketch.
The vertex is at x = −b/2a, and its height is what the discriminant measures indirectly. A parabola opening upward with its vertex below the axis must cross twice; one with its vertex above the axis cannot cross at all. Zero is the exact moment the vertex sits on the axis.
Using it in reverse
Because the discriminant controls the number of roots, questions often run the logic backwards: for what values of k does this equation have exactly one root?
Set the discriminant to zero and solve for k. For x² + kx + 9 = 0 that gives k² − 36 = 0, so k = ±6 — the two values that make the quadratic a perfect square. Asked for two distinct roots instead, solve the inequality k² − 36 > 0.
Questions about discriminant formula
What is the discriminant formula?
D = b² − 4ac, taken from a quadratic in standard form ax² + bx + c = 0. It is the expression under the square root in the quadratic formula.
What does a discriminant of zero mean?
There is exactly one root, repeated. The quadratic is a perfect square and its parabola touches the x-axis at a single point rather than crossing it.
How do I know if a quadratic will factor?
Check whether the discriminant is a perfect square. If it is, the roots are rational and integer factors exist. If not, factoring over the integers is impossible.
Can the discriminant be negative?
Yes, and it simply means there are no real roots. The two roots are complex conjugates, and the parabola never reaches the x-axis.