Solve quadratic equations using the quadratic formula with step-by-step solutions, discriminant analysis, and graphing.
Type the values for a, b, and c in the equation.
Click solve to find the roots of the quadratic equation.
See both roots, discriminant value, and the solution steps.
The roots of ax^2 + bx + c = 0 fall out of one formula: x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Feed the tool the three coefficients and it works through that formula in public, showing the discriminant, each substitution, and the simplified roots, with the parabola drawn and its x-intercepts marked. The most common error is sign handling: when the equation already shows a minus, as in x^2 - 5x + 6 = 0, the middle coefficient is b = -5, not 5. The discriminant alone predicts the outcome: positive means two real roots, zero means one repeated root, negative means two complex roots.
A concrete input and expected output from the current implementation.
Input
Equation: x^2 - 5x + 6 = 0 (a = 1, b = -5, c = 6)
Expected output
Discriminant: D = (-5)^2 - 4(1)(6) = 25 - 24 = 1. Since D > 0 there are two real roots. x = (5 +/- sqrt(1)) / 2, so x = (5 + 1) / 2 = 3 and x = (5 - 1) / 2 = 2. Roots: x = 2 and x = 3. The parabola crosses the x-axis at x = 2 and x = 3, with its vertex at x = 2.5.
D = 1 is a positive perfect square, so the roots are real, distinct, and rational, matching the factorization (x - 2)(x - 3). The vertex lies at x = 2.5, halfway between the roots, because a parabola is symmetric about its axis.
For ax^2 + bx + c = 0 the roots are x = (-b +/- sqrt(b^2 - 4ac)) / (2a). The tool evaluates the discriminant D = b^2 - 4ac first, because its sign fixes the case: D > 0 gives two distinct real roots, D = 0 gives one repeated root with the vertex sitting on the x-axis, and D < 0 gives two complex conjugate roots. It then substitutes and simplifies each branch of the +/- sign separately.
Each stage of the arithmetic appears as its own line: discriminant value, substitution into the formula, simplification. That layout exists so you can follow along or find the exact point where your own working diverged. The graph plots y = ax^2 + bx + c, marks the real roots as x-intercepts, and shows the vertex at x = -b/(2a); the parabola opens upward when a > 0 and downward when a < 0.
The +/- in the formula still produces two values, but when the discriminant is zero both branches give the same number, so the roots coincide. Geometrically the parabola just touches the x-axis at its vertex: one x-intercept, two algebraically equal roots. The two descriptions are the same case seen from different sides.
Move every term to one side first so the equation is in standard form, then enter the coefficients in order with their signs: for 2x^2 - 8x + 6 = 0 that is a = 2, b = -8, c = 6. The tool solves exactly what you give it, so getting the standard form right is the one part it cannot do for you.
The example and behavioral notes were checked against the browser implementation. Standards and primary references below define the relevant format, formula, or platform behavior.
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