Enter the coefficients a, b, and c of a quadratic equation (ax² + bx + c = 0) to compute the discriminant (D = b² − 4ac). You'll see the discriminant value, the nature of the roots (two distinct real roots, one repeated real root, or two complex roots), and the actual roots of the equation. Also try the Cronbach's Alpha Calculator.
Discriminant (D = b² − 4ac)
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Nature of Roots
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Root 1 (x₁)
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Root 2 (x₂)
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The discriminant is the expression D = b² − 4ac derived from the coefficients of a quadratic equation ax² + bx + c = 0. It tells you how many real or complex roots the equation has without needing to fully solve it. The term comes from the Latin word for 'distinguish', because it distinguishes between different types of solutions. See also our Meta-Analysis Calculator.
Given a quadratic equation in standard form ax² + bx + c = 0, identify the coefficients a, b, and c, then substitute them into the formula D = b² − 4ac. For example, for 3x² − 4x + 5 = 0, you get D = (−4)² − 4(3)(5) = 16 − 60 = −44.
If D > 0, the equation has two distinct real roots. If D = 0, there is exactly one real root (a repeated or double root). If D < 0, the equation has two complex (non-real) conjugate roots. This information is useful even before solving the equation.
A multiple root, also called a repeated or double root, occurs when the discriminant equals zero. In this case, both roots are identical: x = −b / (2a). Geometrically, this means the parabola touches the x-axis at exactly one point (the vertex sits on the axis). You might also find our use the Cluster Analysis Calculator useful.
Yes. When D < 0, the square root in the quadratic formula produces an imaginary number, so the two roots are complex conjugates of the form x = (−b ± i√|D|) / (2a). The parabola does not cross or touch the x-axis in this case.
The coefficient a determines whether the equation is truly quadratic. If a = 0, the equation becomes linear (bx + c = 0) and the discriminant formula no longer applies. Always ensure a ≠ 0 when using this calculator.
The quadratic formula is x = (−b ± √D) / (2a), where D = b² − 4ac is the discriminant. The ± symbol is what produces two roots. If D is positive you take two real square roots; if D is zero both roots collapse to one; if D is negative the square root is imaginary.
Yes. You can enter any real number — positive, negative, or decimal — for the coefficients a, b, and c. The calculator computes the exact discriminant and shows the roots in decimal form. Just make sure the coefficient a is not zero.