Quadratic Equation Solver

Delta = b^2 - 4ac. Delta > 0: two real roots; Delta = 0: one repeated real root; Delta < 0: complex conjugate pair.

Decimal is convenient for follow-up calculations, but simplified radical form ((3 + sqrt(5))/2) is exact and often required in algebra classes.

The vertex (-b/2a, c - b^2/4a) is marked amber; real roots green; y-intercept red — all key features at a glance.

Yes, the Quadratic Equation Solver is 100% free with no registration, no hidden fees, and no usage limits. All processing happens locally in your browser, ensuring complete privacy.

Absolutely. The Quadratic Equation Solver processes everything client-side in your browser. No data is uploaded to or stored on any server. Your content remains private on your device at all times.

Yes, the Quadratic Equation Solver is fully responsive and works on smartphones and tablets. You can use it on any device with a modern web browser -- no app download required.

No account or registration is needed. Simply open the Quadratic Equation Solver in your browser and start using it immediately. There are no sign-up walls or usage restrictions.

The Quadratic Equation Solver uses industry-standard formulas and algorithms to ensure accurate results. However, for critical financial or medical decisions, always consult a qualified professional.

The Quadratic Equation Solver works in all modern browsers including Chrome, Firefox, Safari, Edge, and Opera. For the best experience, use the latest version of your preferred browser.

First write the equation in standard form, ax^2 + bx + c = 0, then identify the three coefficients a, b, and c, where a cannot be zero. Substitute them into the quadratic formula, x = (-b +/- sqrt(b^2 - 4ac)) / (2a). Compute the discriminant b^2 - 4ac under the square root first, take its root, then evaluate the plus and minus cases separately to get both solutions. The hardest part is keeping signs and arithmetic straight, especially with negative coefficients. This solver reproduces every one of those stages in its Steps tab: it writes the equation, lists a, b, and c, computes the discriminant, substitutes into the formula, and simplifies. Enter your coefficients above to watch the full substitution and check your own working against it line by line.

Yes. When the discriminant b^2 - 4ac is negative, the equation has no real roots because you would need the square root of a negative number, which does not exist among the real numbers. Geometrically, the parabola never crosses the x-axis, so there are no x-intercepts. This matters in word problems that only accept real answers, such as a length or a time, where a negative discriminant tells you the situation is impossible. The equation still has two solutions in the complex numbers, written as a conjugate pair p +/- qi. This solver detects all three cases automatically and, when the discriminant is negative, displays the complex roots in p +/- qi form rather than just failing. Type your coefficients above and the tool reports the discriminant and the exact nature of the roots instantly.

All three forms describe the same parabola but highlight different features. Standard form, ax^2 + bx + c, makes the y-intercept obvious. Factored form, a(x - r1)(x - r2), exposes the roots directly, since each factor equals zero at a root, so it only exists cleanly when the equation has real roots. Vertex form, a(x - h)^2 + k, reveals the vertex (h, k), the turning point and axis of symmetry, which is ideal for graphing and finding maximum or minimum values. Converting by hand means completing the square or factoring, both error-prone. This solver rewrites your equation in factored and vertex form automatically, marking when an equation is not factorable over the reals. Enter a, b, and c above to see all three representations side by side.

The vertex is the parabola's turning point, and its x-coordinate is always h = -b / (2a). Substitute that value back into the equation to get the y-coordinate k, giving the vertex (h, k). The axis of symmetry is the vertical line x = h that passes through the vertex and mirrors the curve, so the two roots, when real, sit at equal distances on either side of it. Whether the vertex is a minimum or a maximum depends on the sign of a: a positive a opens upward to a minimum, a negative a opens downward to a maximum. This solver computes the vertex, axis of symmetry, opening direction, domain, and range for you, and marks the vertex in amber on its graph. Enter your coefficients above to get every parabola property at once.