What happens if 'a' is zero?

If a = 0, the equation is no longer quadratic — it becomes linear (bx + c = 0), solved as x = −c/b. The quadratic formula is undefined when a = 0 because it divides by 2a.

What is the axis of symmetry?

The vertical line x = −b / 2a passing through the parabola's vertex. It is also the midpoint between the two roots whenever they are real and distinct.

What do complex roots mean geometrically?

Complex roots mean the parabola does not intersect the x-axis. The vertex is above the x-axis (if a > 0) or below it (if a < 0). The roots exist in the complex plane but not on the real number line.

How can I verify my roots?

Substitute each root back into the original equation ax² + bx + c. The result should equal 0 (or very close to 0 due to rounding). This confirms the solution is correct.

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IPL 2026

Quadratic Formula Calculator

Solve quadratic equations (ax² + bx + c = 0) instantly with full working. Shows the discriminant, root type classification, and both roots — real or complex. Useful for algebra, physics projectile problems, and parabola analysis.

Complete User Guide

Enter the three coefficients: a (for x²), b (for x), and c (the constant). The calculator applies x = [−b ± √(b² − 4ac)] / 2a, computes the discriminant, and shows both roots. If the discriminant is negative, roots are shown in complex form a ± bi.

The Mathematical Formula

x = (-b ± √(b² - 4ac)) / 2a

The discriminant Δ = b² − 4ac determines root type. Δ > 0: two distinct real roots. Δ = 0: one repeated root (vertex touches x-axis). Δ < 0: two complex conjugate roots (parabola doesn't cross x-axis). The vertex of the parabola is always at x = −b / 2a.

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