calculator with quadratic formula

What this calculator does

A quadratic equation has the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. This calculator uses the quadratic formula to find the roots (solutions for x) and also shows the discriminant so you can understand the type of roots you’re getting.

How the quadratic formula works

The formula x = (-b ± √(b² - 4ac)) / (2a) gives up to two roots because of the ± sign: one using plus, and one using minus.

• If the discriminant is positive, you get two distinct real roots.
• If the discriminant is zero, both roots are equal (a repeated real root).
• If the discriminant is negative, roots are complex conjugates.

Understanding the discriminant

The discriminant is the expression under the square root: Δ = b² - 4ac. It is one of the most useful checks in algebra because it tells you immediately what kind of answer to expect.

Quick interpretation guide

• Δ > 0: Two real roots.
• Δ = 0: One real root (double root).
• Δ < 0: Two complex roots.

Worked examples

Example 1: x² - 5x + 6 = 0

Here, a = 1, b = -5, c = 6. Discriminant: Δ = (-5)² - 4(1)(6) = 25 - 24 = 1. Since Δ > 0, there are two real roots: x = (5 ± 1)/2, so x = 3 and x = 2.

Example 2: 2x² + 4x + 2 = 0

a = 2, b = 4, c = 2. Δ = 4² - 4(2)(2) = 16 - 16 = 0. One repeated root: x = -4/(2·2) = -1.

Example 3: x² + 2x + 5 = 0

a = 1, b = 2, c = 5. Δ = 2² - 4(1)(5) = 4 - 20 = -16. Because the discriminant is negative, roots are complex: x = -1 ± 2i.

Common mistakes to avoid

• Forgetting that b is part of -b in the formula.
• Dropping parentheses with negative values (especially for b and c).
• Confusing b² with (-b)² in your mental arithmetic.
• Ignoring the ± and only calculating one root.

When to use this calculator

This tool is useful for algebra homework, SAT/ACT prep, engineering calculations, physics trajectory problems, optimization models, and financial growth/decay approximations that reduce to a quadratic form. It is also helpful for checking hand-calculated work quickly.

Final thoughts

The quadratic formula is one of the most dependable formulas in mathematics. Once you know how to identify a, b, and c and evaluate the discriminant, solving quadratics becomes straightforward. Use the calculator above to speed up your workflow, then review the steps to strengthen your understanding.