Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 and get the roots, discriminant, and vertex — runs 100% in your browser with no upload.
Equation: x² − 3x + 2 = 0 Discriminant: 1 Nature: Two distinct real roots Roots: 2, 1 Vertex: (1.5, -0.25) Sum of roots: 3 Product of roots: 2
🔒 Solved instantly in your browser — nothing is uploaded.
What this quadratic equation solver does
Enter the three coefficients a, b, and c and this tool instantly solves the quadratic equation ax² + bx + c = 0. It shows the discriminant, the nature of the roots, the roots themselves — real or complex — and the vertex of the parabola. When the leading coefficient a is zero the equation is no longer quadratic, so the tool falls back to solving the linear equation bx + c = 0, and it clearly flags the special cases of no solution or infinitely many solutions.
How the roots are computed
The solver uses the classic quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The value under the square root, b² − 4ac, is the discriminant: a positive discriminant gives two distinct real roots, a zero discriminant gives one repeated real root, and a negative discriminant gives a pair of complex conjugate roots written in the form real ± imaginary·i. The vertex is reported as the point (−b/2a, f(−b/2a)), the turning point of the parabola. Everything recalculates live as you type.
Frequently asked questions
What happens if I set a to 0?
The equation stops being quadratic, so the tool solves the linear equation bx + c = 0 instead. If both a and b are zero it reports either no solution or infinitely many solutions depending on the constant c.
Is anything I type uploaded to a server?
No. Every calculation runs locally in your browser with JavaScript. The coefficients you enter are never sent, stored, or logged anywhere.
How are complex roots shown?
When the discriminant is negative the two roots are complex conjugates and are displayed in the form 'real + imaginary·i' and 'real − imaginary·i', for example -1 + 2i and -1 - 2i.
What are the sum and product of the roots?
By Vieta's formulas the sum of the roots is −b/a and the product is c/a — both shown alongside the answer. They are always real, even when the roots themselves are complex, and give a quick way to check your work: for x² − 3x + 2 the sum is 3 and the product is 2.