Quadratic Equation Solver – Find Roots of ax² + bx + c = 0

Solve any quadratic equation instantly with our free online quadratic equation solver. Find real and complex roots using the quadratic formula with detailed step-by-step solutions.

Equation Format:

ax² + bx + c = 0

Understanding Quadratic Equations

A quadratic equation is any equation that can be written in the form ax² + bx + c = 0, where a ≠ 0. The name comes from "quadratus," Latin for square — the variable is squared.

The Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a)

The formula gives you both roots at once. The ± symbol means you calculate two values: one with addition and one with subtraction. The expression under the square root, b² - 4ac, is called the discriminant.

What the Discriminant Tells You

Δ > 0

Two distinct real roots. The parabola crosses the x-axis at two points.

Δ = 0

One repeated real root. The parabola just touches the x-axis at its vertex.

Δ < 0

Two complex conjugate roots. The parabola never touches the x-axis.

Worked Examples

Example 1: x² - 5x + 6 = 0

Here a=1, b=-5, c=6. The discriminant is Δ = 25 - 24 = 1.

x = (5 ± √1) / 2 = (5 ± 1) / 2
x₁ = 6/2 = 3
x₂ = 4/2 = 2

Two real roots: x = 3 and x = 2. You can verify: (3)² - 5(3) + 6 = 0 ✓

Example 2: x² + 2x + 1 = 0

Here a=1, b=2, c=1. The discriminant is Δ = 4 - 4 = 0.

x = (-2 ± √0) / 2 = -2/2 = -1

One repeated root: x = -1. This factors as (x+1)² = 0.

Example 3: x² + 2x + 5 = 0

Here a=1, b=2, c=5. The discriminant is Δ = 4 - 20 = -16.

x = (-2 ± √(-16)) / 2 = (-2 ± 4i) / 2
x₁ = -1 + 2i
x₂ = -1 - 2i

Two complex roots. The parabola y = x² + 2x + 5 never crosses the x-axis.

A Quick Fact

The quadratic formula in its modern form was developed by the Persian mathematician Al-Khwarizmi around 820 CE. His book "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala" (The Compendious Book on Calculation by Completion and Balancing) gave us the word "algebra."

Common Questions

What if a = 0?

Then it's not a quadratic equation anymore — it becomes linear: bx + c = 0. You'd solve it as x = -c/b. This calculator requires a ≠ 0 to use the quadratic formula.

Can I use this for equations not set to zero?

First rearrange so one side equals zero. For example, x² + 3x = 10 becomes x² + 3x - 10 = 0. Then a=1, b=3, c=-10.

Why do complex roots come in pairs?

For polynomials with real coefficients, complex roots always appear as conjugate pairs (a + bi and a - bi). This ensures the coefficients stay real when you expand (x - r₁)(x - r₂).

When would I need complex solutions?

In electrical engineering, complex numbers describe AC circuits. In physics, they appear in quantum mechanics and wave equations. The complex roots aren't "fake" — they represent real phenomena.

Is the quadratic formula the only way to solve?

No. You can also factor (when possible), complete the square, or graph and find x-intercepts. The quadratic formula always works, but factoring is faster when the equation factors nicely.

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