Discriminant
Compute the discriminant Delta = b^2 - 4ac for any quadratic ax^2+bx+c. Determine whether roots are real and distinct, real and equal, or complex with a clear explanation.
About This Calculator
The Discriminant Calculator computes the discriminant Delta = b^2 - 4ac for any quadratic equation ax^2 + bx + c = 0. The discriminant is the expression under the square root in the quadratic formula and is the single most important value for understanding the nature of a quadratic's roots without actually solving it.
A positive discriminant (Delta > 0) means two distinct real roots -- the parabola crosses the x-axis at two points. A zero discriminant (Delta = 0) means one repeated real root -- the parabola just touches the x-axis at its vertex. A negative discriminant (Delta < 0) means two complex conjugate roots -- the parabola never intersects the x-axis. This is known as the discriminant's "nature of roots" classification.
The discriminant also appears in higher-degree polynomials and conic sections. In quadratic equations, it is also used to check whether a quadratic can be factored over the rational numbers (Delta being a perfect square), and it determines the number of real solutions in quadratic optimization problems.
Frequently Asked Questions
What is the discriminant formula?
The discriminant Delta = b^2 - 4ac, where a, b, c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
What does a positive discriminant mean?
If Delta > 0, the quadratic has two distinct real roots. The parabola crosses the x-axis at two different points. If Delta is a perfect square, the roots are rational; otherwise they are irrational.
What does a zero discriminant mean?
If Delta = 0, the quadratic has exactly one repeated real root (a double root). The parabola touches the x-axis at exactly one point (its vertex lies on the x-axis).
What does a negative discriminant mean?
If Delta < 0, the quadratic has two complex conjugate roots with no real solutions. The parabola does not cross the x-axis at all.
How is the discriminant related to the quadratic formula?
The discriminant is the expression under the square root in the quadratic formula x = (-b +/- sqrt(b^2 - 4ac)) / 2a. Its sign determines whether the square root yields a real or imaginary number.
What is the discriminant of a cubic equation?
For cubic equations ax^3 + bx^2 + cx + d = 0, the discriminant is more complex: Delta = 18abcd - 4b^3d + b^2c^2 - 4ac^3 - 27a^2d^2. It determines whether roots are all real or one real and two complex.
Can the discriminant be used in real-world problems?
Yes, the discriminant is used in physics (projectile motion), engineering (control systems), finance (quadratic profit models), and computer graphics (ray intersection testing).
Is this calculator free?
Yes, all calculators on Calculy are completely free to use with no limits or registration required.