Solve ax² + bx + c = 0 with step-by-step solutions and graph
Parabola Vertex Roots
The Quadratic Solver finds the real or complex roots of any second-degree equation written in the standard form ax^2 + bx + c = 0. Enter the three coefficients a, b, and c (a must be non-zero), and the tool returns the exact and decimal solutions, the discriminant, the vertex, and a plotted parabola so you can see where the curve crosses the x-axis.
It is built for students working through algebra homework, teachers checking answers, and anyone who needs to factor or graph a quadratic quickly. Because it shows each step — computing the discriminant, applying the quadratic formula, and simplifying — it doubles as a learning aid rather than just a black-box answer.
Every quadratic ax^2 + bx + c = 0 (with a not equal to 0) is solved by the quadratic formula: x = (-b +/- sqrt(b^2 - 4ac)) / (2a). The term inside the square root, b^2 - 4ac, is called the discriminant and it decides how many real solutions exist.
When the discriminant is positive, there are two distinct real roots and the parabola crosses the x-axis at two points. When it equals zero, there is one repeated real root and the vertex sits exactly on the x-axis. When it is negative, the square root of a negative number produces two complex conjugate roots of the form p +/- qi, and the parabola never touches the x-axis.
As a worked example, for 2x^2 - 4x - 6 = 0 the discriminant is (-4)^2 - 4(2)(-6) = 16 + 48 = 64. Taking the square root gives 8, so x = (4 +/- 8) / 4, yielding x = 3 and x = -1. The tool also reports the vertex at x = -b/(2a) = 1, y = -8, which is the parabola's lowest point.
The discriminant b^2 - 4ac tells you the nature of the roots without fully solving. Positive means two real roots, zero means one repeated real root, and negative means two complex (non-real) roots.
Yes. When the discriminant is negative, the tool returns the two complex conjugate roots in the form p + qi and p - qi instead of failing, and the graph shows a parabola that stays entirely above or below the x-axis.
If a = 0 the x^2 term disappears and the equation becomes linear (bx + c = 0), not quadratic. The quadratic formula divides by 2a, so a = 0 would be undefined. Enter a non-zero value for a.
Yes. You can enter negative numbers, fractions written as decimals, and non-integer coefficients. The solver applies the same formula and returns decimal results, rounding for display while keeping precision in the calculation.
The vertex is the turning point of the parabola, located at x = -b/(2a). It marks the minimum value when a is positive or the maximum value when a is negative, and it is the axis of symmetry for the two roots.
Yes, it is completely free with no sign-up. All calculations run locally in your browser, so the coefficients you enter are never uploaded to a server or stored anywhere.
Absolutely. Because it shows the discriminant, the substituted quadratic formula, and each simplification step, you can compare your own working line by line rather than just checking the final answer.