Quadratic Equation Calculator
Solve ax² + bx + c = 0
Enter the coefficients of any quadratic equation to instantly get the roots, discriminant, vertex, sum & product of roots — with full working steps and a live graph of the parabola.
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How to Solve a Quadratic Equation
A quadratic equation has the standard form ax² + bx + c = 0, where a, b and c are constants and a ≠ 0. The roots are found using the quadratic formula:
x = (−b ± √(b² − 4ac)) / 2a
The discriminant (b² − 4ac) tells you the nature of the roots before you even calculate them: if it's positive, there are two distinct real roots; if it's zero, there's one repeated real root; and if it's negative, the roots are complex (involving √−1, written as i).
This calculator also shows the vertex of the parabola (the minimum or maximum point), and the sum and product of roots — two shortcuts (−b/a and c/a respectively) that are frequently tested in board exams and JEE/NEET without needing to solve the full equation.
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A negative discriminant means the equation has no real roots — the parabola never crosses the x-axis. The roots are complex numbers, written in the form p ± qi, where i = √−1. This calculator shows both the real and imaginary parts.
If a = 0, the equation is no longer quadratic — it becomes a linear equation (bx + c = 0). The calculator will still solve it for you, but a must be non-zero for it to be a true quadratic equation.
The vertex is the turning point of the parabola — the minimum point if a is positive, or the maximum point if a is negative. It's calculated at x = −b/2a, with the y-value found by substituting that x back into the equation.
Sum of roots (−b/a) and product of roots (c/a) let you answer many exam questions — like forming a new equation from given roots, or checking your answer — without fully solving the quadratic formula each time.