Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 instantly: real or complex roots, the discriminant, the vertex and full step-by-step working. Free and private — solved entirely in your browser.
How to use the quadratic equation solver
- Write your equation as ax² + bx + c = 0. Move every term to the left side first, so 2x² = 4x + 6 becomes 2x² − 4x − 6 = 0.
- Enter a, b and c. Decimals and negative numbers are fine. The equation preview confirms what you typed and the roots update instantly.
- Read the solution. You get the discriminant, both roots (real or complex), the vertex, and the full quadratic-formula working you can copy for homework.
About this tool
This free quadratic equation solver applies the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The discriminant D = b² − 4ac decides the nature of the roots: D > 0 gives two distinct real roots, D = 0 gives one repeated real root, and D < 0 gives a pair of complex conjugate roots, which the solver writes out in a + bi form rather than just saying "no solution". It also reports Vieta's formulas (the sum −b/a and product c/a of the roots) and the parabola's vertex, which are useful for checking answers and sketching the graph.
The solver uses a numerically stable variant of the quadratic formula to avoid the rounding errors the textbook formula suffers from when b is much larger than a and c. Everything runs locally in your browser — no equation you type is ever uploaded or stored. If a = 0, the equation is no longer quadratic; the tool detects this and solves the resulting linear equation bx + c = 0 instead.
Frequently asked questions
What is the discriminant and what does it tell me?
The discriminant is D = b² − 4ac, the expression under the square root in the quadratic formula. If D is positive there are two different real roots; if it is exactly zero the parabola touches the x-axis at one repeated root; if it is negative the roots are complex conjugates and the parabola never crosses the x-axis.
Can this solver find complex (imaginary) roots?
Yes. When D < 0 the roots are x = −b/2a ± i·√(−D)/2a, and the solver prints them in standard a + bi form, for example x₁ = 1 + 2i and x₂ = 1 − 2i. Complex roots of a real quadratic always come as a conjugate pair.
What happens if a is 0?
With a = 0 there is no x² term, so the equation is linear: bx + c = 0. The solver tells you this and returns x = −c/b when b ≠ 0. If both a and b are 0, the equation is either always true (c = 0) or has no solution (c ≠ 0).
How accurate are the results?
Coefficients are handled as double-precision floating-point numbers and the solver uses a cancellation-free form of the quadratic formula, so results are typically accurate to about 15 significant digits. Displayed values are rounded to 8 significant digits; copy the steps for the values shown.