The discriminant Δ = b²−4ac determines the nature of roots: Δ>0 → two distinct real roots; Δ=0 → double root; Δ<0 → no real roots.
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Solve ax²+bx+c=0 with the quadratic formula, discriminant and real or complex roots.
The quadratic equation ax² + bx + c = 0 (with a ≠ 0) is solved by the quadratic formula: x = (−b ± √Δ) ÷ 2a, where Δ = b² − 4ac is the discriminant. It is the discriminant that reveals the nature of the roots before you even compute them: if Δ > 0, there are two distinct real roots; if Δ = 0, there is a single (double) real root at x = −b ÷ 2a; if Δ < 0, there is no real solution, and the roots are complex conjugates of the form −b/(2a) ± (√|Δ| / 2|a|)·i. This tool takes the three coefficients, computes Δ, classifies the case and shows the full step-by-step, from assembling the equation to the final x₁ and x₂ values. Coefficients b and c may be zero (allowing incomplete equations like x² − 5 = 0), but a cannot, since it would cancel the quadratic term. The computation uses 64-bit floating point, with full precision for coefficients whose magnitude is below 10¹⁵, and displays up to 10 significant digits. Everything runs locally in your browser.
The formula x = (−b ± √Δ) ÷ 2a solves any real quadratic equation and comes from completing the square in ax² + bx + c = 0. All of the roots' behavior is concentrated in the discriminant Δ = b² − 4ac, the number under the root: since you cannot take a real square root of a negative number, the sign of Δ decides everything.
There are three scenarios. Δ > 0: the root is a positive real number and the ± sign produces two distinct real roots. Δ = 0: the root is zero, the ± disappears and a single double root at −b ÷ 2a remains. Δ < 0: the root would be of a negative, so there is no real solution and the tool expresses the two complex conjugate roots.
Example 1, two real roots: x² − 5x + 6 = 0 (a = 1, b = −5, c = 6). The discriminant is Δ = (−5)² − 4·1·6 = 25 − 24 = 1, positive. Since √1 = 1, x = (5 ± 1) ÷ 2, giving x₁ = 3 and x₂ = 2. Check: 3 + 2 = 5 (= −b/a) and 3 × 2 = 6 (= c/a).
Example 2, complex roots: x² + 2x + 5 = 0 (a = 1, b = 2, c = 5). The discriminant is Δ = 2² − 4·1·5 = 4 − 20 = −16, negative. The real part is −b ÷ 2a = −1, and the imaginary part is √16 ÷ 2 = 2, so the roots are −1 + 2i and −1 − 2i.
There are two quick ways to check an answer. The first is to substitute the root back into the original equation: ax² + bx + c should equal zero (or be very close, due to floating-point rounding). The second is Vieta's formulas: the sum of the roots is −b ÷ a and the product is c ÷ a, in Example 1, sum 5 and product 6, exactly as expected. Be careful with the sign when typing negative b and c, and remember a cannot be zero.
Quadratic equations appear in physics (motion under constant acceleration, a projectile's height), in area and geometry problems, in optimization (the vertex of the parabola gives the maximum or minimum) and in finance and engineering. Classifying by Δ tells you at once whether a problem has a real solution.
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The discriminant Δ = b²−4ac determines the nature of roots: Δ>0 → two distinct real roots; Δ=0 → double root; Δ<0 → no real roots.
ax² + bx + c = 0All calculations stay in your browser. No data is sent to any server.