Quadratic Formula Calculator

What Is the Quadratic Formula?

The quadratic formula is a closed-form algebraic solution for any equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0. It gives the value(s) of x that satisfy the equation using the expression:

x = (−b ± √(b² − 4ac)) / 2a

This formula is derived by completing the square on the general quadratic equation. It is one of the most frequently used algebraic tools across mathematics, physics, engineering, economics, and data science — any field that involves parabolic relationships, optimization, or root-finding.

How to Use This Calculator

1. Enter coefficient a — the quadratic coefficient (cannot be zero). If a = 0, the equation is linear, not quadratic.
2. Enter coefficient b — the linear coefficient. Can be positive, negative, or zero.
3. Enter coefficient c — the constant term. Can be positive, negative, or zero.
4. Read the result — the calculator displays the root(s) in real or complex form, including the discriminant's effect on the solution type.

The inputs accept decimals, negative values, and fractions (entered as decimals). The calculator automatically updates as you type — no button press required.

What the Discriminant Tells You

The discriminant is the part under the square root: D = b² − 4ac. Its sign determines the nature of the roots before you finish calculating:

• D > 0 — two distinct real roots. The parabola crosses the x-axis at two points. This is the most common case in applied math.
• D = 0 — one repeated real root (also called a double root). The parabola touches the x-axis at exactly one point (its vertex).
• D < 0 — two complex conjugate roots (involving i, where i² = −1). The parabola never crosses the x-axis; the equation has no real-number solution.

Worked Example: 2x² + 5x − 3 = 0

Using the coefficients a = 2, b = 5, c = −3 from the default inputs:

1. Calculate the discriminant: D = 5² − 4(2)(−3) = 25 + 24 = 49
2. Take the square root: √49 = 7
3. Apply the formula: x = (−5 ± 7) / (2 × 2) = (−5 ± 7) / 4
4. Solve both cases: x = (−5 + 7)/4 = 2/4 = 0.5 and x = (−5 − 7)/4 = −12/4 = −3

The roots are x = 0.5 and x = −3. You can verify by plugging either value back into the original equation — both will satisfy 2x² + 5x − 3 = 0.

Common Use Cases

• Physics: Solving projectile motion equations (height vs. time) to find when an object hits the ground.
• Engineering: Finding natural frequencies, breakpoints, and stability thresholds in control systems.
• Finance: Solving for breakeven points in quadratic revenue/cost models.
• Data Science: Computing closed-form solutions for least-squares regression and optimization problems.
• Algebra/Education: Checking homework answers and understanding the relationship between coefficients and roots.

Frequently Asked Questions