The discriminant is the b² − 4ac part of the quadratic formula. Its sign tells you how many solutions to expect before you solve anything.
What is the discriminant?
The discriminant of a quadratic ax² + bx + c is the number b² − 4ac. It tells you how many real solutions the equation ax² + bx + c = 0 has, and what kind they are, without solving it.
It's the expression under the square root in the quadratic formula. Whatever happens to that square root (positive, zero, or negative inside) decides what the roots look like.
Some books use the Greek letter Δ for it instead of D. The name comes from the idea of discriminating, or telling apart: it sorts quadratics by the kind of roots they have.
How do you find the discriminant?
Write the quadratic in standard form, read off a, b, and c, and substitute them into b² − 4ac. Keep the signs attached to each coefficient.
For 2x² + 3x − 5, a = 2, b = 3, and c = −5.
What does the discriminant tell you?
The sign of the discriminant tells you the number of real roots, and with whole-number coefficients, whether those roots are rational. Here are all the cases in one place.
What do the different discriminant cases look like?
Here is one example of each case, so you can see the pattern with real numbers.
x² − 2x − 4: D = (−2)² − 4(1)(−4) = 4 + 16 = 20. That is positive but not a perfect square, so the roots are irrational: x = 1 + √5 and x = 1 − √5.
x² − 6x + 9: D = (−6)² − 4(1)(9) = 36 − 36 = 0. There is one repeated root, x = 3. The trinomial is a perfect square, (x − 3)².
x² + 2x + 5: D = 2² − 4(1)(5) = 4 − 20 = −16. There are no real roots. The complex roots are x = −1 + 2i and x = −1 − 2i.
What does the discriminant mean on a graph?
The discriminant counts the x-intercepts of the parabola y = ax² + bx + c. Two real roots means two x-intercepts, one root means the vertex sits on the x-axis, and no real roots means the graph stays above or below the axis.
For y = x² + 2x + 5, the parabola opens up and its lowest point is (−1, 4). It never comes down to y = 0, which matches D = −16. For y = x² − 6x + 9, the vertex is (3, 0), right on the axis, which matches D = 0. You can see each case for yourself with the graphing calculator.
How can the discriminant tell you if a quadratic factors?
If a, b, and c are integers, the quadratic factors over the integers exactly when D is a perfect square (0, 1, 4, 9, 16, 25, and so on). That makes the discriminant a quick test before you spend time hunting for factor pairs.
For x² + 3x − 2, D = 9 + 8 = 17. Since 17 isn't a perfect square, no pair of integers will work, and you should go straight to the quadratic formula. For 2x² + 3x − 5, D = 49 = 7², and it does factor: (2x + 5)(x − 1).
Why does the discriminant work?
It works because the quadratic formula takes the square root of b² − 4ac. A positive number has two real square roots, zero has one, and a negative number has none among the real numbers.
So when D > 0, the ± in x = (−b ± √D)/2a gives two different answers. When D = 0, adding and subtracting zero gives the same answer twice, x = −b/2a. When D < 0, √D is imaginary and both answers are complex.
There's a graph reason as well. The vertex of y = ax² + bx + c sits at height −D/4a. If a > 0 and D > 0, that height is negative, so the parabola dips below the x-axis and has to cross it twice on the way up.
How do you use the discriminant to find an unknown coefficient?
Set the discriminant to match the condition in the question, then solve for the unknown. One solution means D = 0, two real solutions means D > 0, and no real solutions means D < 0.
Suppose x² + kx + 9 = 0 has exactly one solution. The discriminant is k² − 4(1)(9) = k² − 36, and it must equal 0.
What mistakes do people make with the discriminant?
Most wrong answers come from signs. Watch for these.
- Squaring a negative b incorrectly. (−6)² is 36, not −36. Put b in parentheses before you square it.
- Dropping the sign of c. With c = −5, −4ac becomes −4(2)(−5) = +40.
- Using the equation before it's in standard form. For x² = 4x − 1, rewrite it as x² − 4x + 1 = 0 first.
- Taking the square root of D. The discriminant is b² − 4ac itself, not √(b² − 4ac).
When is the discriminant useful?
Use it when a question asks how many solutions or x-intercepts there are, or what type of roots to expect. It also helps you choose a method: a perfect-square D points to factoring, and any other positive D points to the quadratic formula.
It's handy for word problems too. If a projectile height equation gives a negative discriminant when you set it equal to some height, the object never reaches that height. The quadratic formula complete guide shows what to do once you know a real answer exists.
- Write the equation as ax² + bx + c = 0.
- Compute D = b² − 4ac.
- If D < 0, there are no real solutions, so you can stop.
- If D is a perfect square, factor the quadratic.
- Otherwise, use the quadratic formula and simplify √D.
Frequently asked questions
What is the discriminant formula?
For ax² + bx + c, the discriminant is D = b² − 4ac. It is the part of the quadratic formula under the square root sign.
What does a negative discriminant mean?
A negative discriminant means the quadratic has no real roots. The parabola does not touch the x-axis, and the two solutions are complex conjugates such as −1 ± 2i.
What does a discriminant of zero mean?
It means the quadratic has exactly one real root, repeated twice. The parabola's vertex lies on the x-axis, and the trinomial is a perfect square such as (x − 3)².
Can the discriminant be a fraction?
Yes, if any of a, b, or c are fractions. The sign rules still apply. To use the perfect-square test, multiply through by a common denominator first so the coefficients are integers.
Do you need the discriminant if you use the quadratic formula anyway?
You don't have to compute it separately, since it sits inside the formula. Working it out first still helps, because it tells you whether to expect two answers, one, or none, and whether the square root will come out as a whole number.
Is the discriminant the same as the vertex?
No. The discriminant is a single number that tells you about the roots. The vertex is a point on the graph, found with x = −b/2a. They are related, since the vertex y value is −D/4a.