Quadratic Formula Calculator

About this tool

The Quadratic Formula Calculator finds the roots of any quadratic equation in the form ax² + bx + c = 0 using the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. The tool calculates the discriminant (b² - 4ac) and determines whether the equation has two distinct real roots (discriminant > 0), one repeated real root (discriminant = 0), or two complex conjugate roots (discriminant < 0). As you type, it rewrites your equation, shows the discriminant with your coefficients substituted, then the quadratic formula with your numbers, and finally the roots as decimals (up to 8 significant digits) or as a ± bi for complex roots. If a = 0 it solves the resulting linear equation instead. The quadratic formula is one of the oldest closed-form algebraic results, attributable to Babylonian and Indian mathematicians millennia before its modern symbolic form.

Why use it

• Handles all three discriminant cases: two real roots, one root, and complex roots.

• Step-by-step working helps students follow the quadratic formula.

• Live equation preview catches sign mistakes before you rely on the answer.

• Instant calculation — no form submission needed.

• Falls back to solving bx + c = 0 when a is 0, so linear inputs still get an answer.

• Privacy-first: coefficients are processed entirely in your browser.

How to use

1. Enter the coefficients a, b, and c in the respective fields.
2. Check the equation preview above the fields to confirm the signs are right.
3. Read the step-by-step box: the discriminant with your values, then the quadratic formula with your numbers.
4. Read the roots below — two real roots, one repeated root, or a complex pair a ± bi.
5. Verify the result by substituting each root back into ax² + bx + c — both should yield 0 (within floating-point tolerance).
6. Use the discriminant first if you only need to know whether real roots exist — full computation is unnecessary for that question.

When it helps

• Solving quadratic equations in algebra coursework.
• Checking hand-calculated roots for errors.
• Finding roots of physics, engineering, or economics quadratic models.
• Learning the quadratic formula with step-by-step examples.
• Solving projectile motion or trajectory problems in physics (x represents time or distance).
• Computing break-even points in business and economics where revenue equals cost.

Examples

Tips

• Always check whether your equation is in standard form (ax² + bx + c = 0) before entering coefficients — moving terms across the equals sign changes signs.
• If the discriminant is a perfect square, the roots are rational and can also be found by factoring — try factoring first for cleaner answers.
• For physics problems (projectile motion, kinematics), the negative root is often physically meaningless (negative time) — pick the positive one.
• When |b| is much larger than |c|, beware floating-point cancellation; the alternative formula x = -2c / (b + sign(b)√(b²-4ac)) gives more accurate small roots.
• The sum of the roots equals -b/a and the product equals c/a — use these as a sanity check after computing.
• Complex roots always come in conjugate pairs (a + bi and a - bi) for real-coefficient quadratics — if one root is complex, the other is its conjugate.

Frequently Asked Questions

Glossary

Quadratic equation

An equation of the form ax² + bx + c = 0 with a ≠ 0. Has at most two solutions (roots).

Discriminant

Δ = b² - 4ac. The expression under the square root in the quadratic formula. Sign determines the nature of the roots.

Quadratic formula

x = (-b ± √(b² - 4ac)) / 2a. Gives both roots of any quadratic; provable by completing the square.

Roots / zeros

Values of x where the quadratic equals zero. Geometrically, where the parabola crosses the x-axis.

Vertex

The maximum or minimum point of the parabola; located at x = -b/2a, y = c - b²/4a.

Complex conjugate

If a + bi is a root of a real-coefficient quadratic, so is a - bi. They mirror across the real axis in the complex plane.

Completing the square

Algebraic technique that rewrites ax² + bx + c as a(x + b/2a)² + (c - b²/4a) — the geometric basis of the quadratic formula.