What does the discriminant tell me?

The discriminant D = b² − 4ac reveals the nature of the roots before you evaluate any square roots. D > 0: two distinct real roots (parabola crosses x-axis twice). D = 0: one repeated real root (parabola touches x-axis at one point). D < 0: two complex conjugate roots (parabola never crosses x-axis).

Where does the quadratic formula come from?

It is derived by completing the square on ax² + bx + c = 0 — a classical algebraic technique known in various forms since Babylonian times and in its modern form since the 16th century. The derivation is a standard high-school exercise and is worth working through once to understand the formula deeply.

Can every quadratic be factored?

Over the complex numbers, yes — every quadratic factors into two linear factors. Over the real numbers, only quadratics with non-negative discriminant factor into real-number factors. Over the integers, only quadratics whose discriminant is a perfect square factor with integer roots.

What are real-world uses of quadratic equations?

Projectile motion in physics (position is quadratic in time), economics (profit maximization problems often reduce to quadratics), architecture (parabolic arches), engineering (stress analysis, optimization), signal processing (chirp signals), and finance (quadratic utility models). Any situation involving squared terms produces a quadratic.

How do I find the vertex of a quadratic?

The vertex (minimum or maximum point) occurs at x = −b / (2a). The y-coordinate of the vertex is c − b² / (4a). If a > 0, the vertex is a minimum; if a < 0, it is a maximum. The vertex form y = a(x − h)² + k makes this immediate.

What are complex conjugate roots?

If a quadratic with real coefficients has complex roots, they always come in conjugate pairs: a + bi and a − bi. This is because the square root of a negative number contributes ±i, and the two signs create conjugate pairs. This pattern reveals itself through the ± in the formula itself.

Can I solve cubic or quartic equations with this?

No — this solver handles only quadratics (degree 2). Cubics (degree 3) have their own formula (Cardano's), quartics (degree 4) have an even more elaborate one (Ferrari's), and quintics (degree 5) generally have no closed-form solution over the radicals (Abel–Ruffini theorem).

Quadratic Equation Solver

Math

Solve ax² + bx + c = 0 with detailed root analysis.

Math

Quadratic Equation Solver

Generated on April 24, 2026

Solves ax² + bx + c = 0

?What is the Quadratic Equation Solver?

A quadratic equation solver finds the roots of any equation in the form ax² + bx + c = 0 using the quadratic formula. It automatically identifies whether the roots are two distinct real numbers, a single repeated real root, or a complex conjugate pair — based on the sign of the discriminant. Quadratic equations appear all over mathematics, physics, engineering, and finance: projectile motion, parabolic curves, break-even analysis, optimization problems, and many real-world models reduce to quadratic form once the unknowns are set up.

The Formula

x = [−b ± √(b² − 4ac)] ÷ (2a). Discriminant D = b² − 4ac.

The quadratic formula is derived by 'completing the square' on ax² + bx + c = 0 — a standard algebraic manipulation that isolates x. The discriminant D = b² − 4ac determines the root type before you even evaluate the square root: D > 0 gives two distinct real roots, D = 0 gives one repeated real root (a double root where the parabola just touches the x-axis), and D < 0 gives two complex conjugate roots (the parabola never crosses the x-axis in the real plane).

Practical Examples

x² − 5x + 6 = 0 gives x = 2 and x = 3 (D = 1 > 0, two real roots) — factorable as (x − 2)(x − 3) = 0.

x² − 4x + 4 = 0 gives x = 2 (D = 0, one double root) — the parabola just touches y = 0 and rises.

x² + 2x + 5 = 0 gives x = −1 ± 2i (D = −16 < 0, complex roots) — no real solutions, but complex roots come in conjugate pairs.

Physics projectile motion: the time for a projectile to return to launch height is found by solving a quadratic in t derived from y = v₀t − ½gt².

Business: break-even analysis where revenue and cost are quadratic in production volume often ends up requiring the quadratic formula.

Architecture: parabolic arches and suspension-bridge cable shapes are quadratic — setting cable sag at specific horizontal distances involves solving quadratics.

Frequently Asked Questions

Then the equation is linear (bx + c = 0), not quadratic. The calculator rejects a = 0 to prevent division by zero in the formula. For a linear equation, solve directly: x = −c / b.

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Quadratic Equation Solver

Quadratic Equation Solver

?What is the Quadratic Equation Solver?

The Formula

Practical Examples

Frequently Asked Questions

What if the coefficient a is zero?

What does the discriminant tell me?

Where does the quadratic formula come from?

Can every quadratic be factored?

What are real-world uses of quadratic equations?

How do I find the vertex of a quadratic?

What are complex conjugate roots?

Can I solve cubic or quartic equations with this?

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