🧮 Quadratic Equation Solver

Quadratic Equation Solver

This tool solves any quadratic equation of the form ax² + bx + c = 0. Enter the three coefficients a, b, and c to instantly compute the roots, discriminant, vertex coordinates, axis of symmetry, and the factored form of the equation.

The quadratic formula x = (-b ± √(b² - 4ac)) / 2a is used to find the roots. The discriminant (b² - 4ac) determines whether the roots are real and distinct, real and repeated, or complex conjugates. The vertex represents the turning point of the parabola.

This solver is useful for students studying algebra, calculus, physics, and engineering. It works with integer and decimal coefficients, and handles both real and complex roots with full precision.

Frequently Asked Questions

What is the discriminant of a quadratic equation?

The discriminant is the value b² - 4ac. It determines the nature of the roots: if positive, there are two distinct real roots; if zero, there is one repeated real root; if negative, there are two complex conjugate roots.

What if the discriminant is negative?

When the discriminant is negative, the quadratic equation has no real roots. Instead, it has two complex conjugate roots of the form (-b ± i√|D|) / 2a, where i is the imaginary unit and |D| is the absolute value of the discriminant.

What is the vertex of a parabola?

The vertex is the highest or lowest point on the parabola defined by y = ax² + bx + c. Its x-coordinate is -b/(2a) and its y-coordinate is found by substituting that x-value back into the equation. If a > 0, the vertex is a minimum; if a < 0, it is a maximum.