Quadratic Equation Solver

Solve ax² + bx + c = 0

About Quadratic Equation Solver

Quadratic Equations

A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are constants and a is not zero. The graph of a quadratic function is a parabola. Quadratic equations appear throughout physics (projectile motion, free fall), engineering (optimization problems), economics (profit maximization), and geometry (areas, dimensions). Finding the roots (x-values where the equation equals zero) is one of the most fundamental algebraic skills.

The Quadratic Formula

x = (-b plus or minus sqrt(b² - 4ac)) / (2a). The discriminant (D = b² - 4ac) determines the nature of roots: D > 0 means two distinct real roots, D = 0 means one repeated real root, D < 0 means two complex conjugate roots. Other solving methods include factoring, completing the square, and graphing.

Additional Information

The vertex form: y = a(x - h)² + k where vertex is at (h, k). h = -b/(2a) and k = f(h). Sum of roots = -b/a. Product of roots = c/a. The axis of symmetry is x = -b/(2a). These relationships (Vieta's formulas) connect roots to coefficients without solving.

Frequently Asked Questions

What if the discriminant is negative?

A negative discriminant means no real roots exist. The parabola does not cross the x-axis. The roots are complex numbers: x = (-b plus or minus i*sqrt(|D|)) / (2a), where i = sqrt(-1).

When should I factor instead of using the formula?

Factor when the equation factors neatly into integers (x² - 5x + 6 = (x-2)(x-3)). The quadratic formula always works but factoring is faster when possible. If coefficients are large or roots are irrational, use the formula.

What is the vertex of a parabola?

The vertex is the highest point (if a < 0) or lowest point (if a > 0) of the parabola. Its x-coordinate is -b/(2a). The vertex represents the maximum or minimum value of the quadratic function.