Discriminant Calculator
Calculate the discriminant of a quadratic equation ax² + bx + c = 0 and instantly find out whether the roots are real, repeated, or complex. Just enter your coefficients and get step-by-step results.
The discriminant is the expression under the square root in the quadratic formula. Its value tells you everything you need to know about the nature of the roots before actually solving the equation.
The Discriminant Formula:
D = b² - 4ac
What the Value Tells You:
- D > 0: Two distinct real roots. The parabola crosses the x-axis at two points. If D is a perfect square, the roots are rational.
- D = 0: Exactly one repeated real root (a double root). The parabola just touches the x-axis at one point. The root is x = -b / (2a).
- D < 0: No real roots. Two complex conjugate roots exist instead. The parabola does not cross the x-axis at all.
Why the Discriminant Matters:
The discriminant saves you time. Before going through the full quadratic formula, a quick D check tells you what kind of answer to expect. This is especially useful in:
- Factoring decisions: if D is a perfect square, the quadratic factors cleanly over the integers
- Graphing: knowing the root count tells you the shape of the parabola relative to the x-axis
- Physics and engineering: quickly determining if a system has real solutions
Connection to the Quadratic Formula:
x = (-b ± sqrt(D)) / (2a)
When D is negative, sqrt(D) involves imaginary numbers, which is why the roots become complex.