Quadratic Formula Calculator

Solve quadratic equations ax² + bx + c = 0 with detailed step-by-step solutions.

Enter Coefficients

For the equation ax² + bx + c = 0

a (coefficient of x²) — must not be 0

b (coefficient of x)

c (constant term)

Quick Presets

Roots

x = −2, x = −3

Δ = 1 (two distinct real roots)

Step-by-Step Solution

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Quadratic Formula Calculator: Solve Any Quadratic Equation with Steps

The Quadratic Formula Calculator on lcmcalculator.info is a free online tool that solves quadratic equations of the form ax² + bx + c = 0 using the quadratic formula. Whether you need a quadratic formula calculator with steps for homework, a quadratic equation solver for quick checks, or a quadratic roots calculator for teaching, this tool delivers fast, accurate results with full step-by-step explanations — including real and complex roots.

What Is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2, written in the standard form:

ax² + bx + c = 0, where a ≠ 0

Here, a, b, and c are coefficients (real numbers), and x is the unknown variable. The condition a ≠ 0 is essential — if a = 0, the equation becomes linear, not quadratic.

Quadratic equations appear everywhere in mathematics, physics, engineering, and finance. They model projectile motion, area problems, optimization, and many other real-world phenomena.

What Is the Quadratic Formula?

The quadratic formula is the most reliable method for solving any quadratic equation. It is derived by completing the square on the general form ax² + bx + c = 0:

x = (−b ± √(b² − 4ac)) / (2a)

The symbol ± means there are generally two solutions — one with addition and one with subtraction. These solutions are called the roots of the quadratic equation.

What Is the Discriminant?

The expression under the square root, Δ = b² − 4ac, is called the discriminant. It determines the nature and number of roots:

Δ > 0: Two distinct real roots.
Δ = 0: One repeated real root (a double root).
Δ < 0: Two complex conjugate roots.

Our discriminant calculator (integrated into this tool) automatically computes Δ and explains what it means for your equation.

How to Use the Quadratic Formula Calculator

Using our free quadratic formula solver is straightforward:

Enter the coefficients a, b, and c in the input fields. Remember that a must not be 0.
The calculator updates live and shows the roots, the discriminant, and the full step-by-step solution.
Copy the solution using the button provided.

Step-by-Step Solution Process

Our quadratic formula calculator with steps shows each part of the process:

Identify coefficients: Write down a, b, and c.
Compute the discriminant: Δ = b² − 4ac.
Determine the nature of roots: Based on Δ.
Apply the quadratic formula: x = (−b ± √Δ) / (2a).
Simplify: Compute the two roots.
Verify: Optional check by substituting back.

Worked Examples

Example 1: x² + 5x + 6 = 0

Coefficients: a = 1, b = 5, c = 6.

Discriminant: Δ = 5² − 4×1×6 = 25 − 24 = 1.

Since Δ > 0, there are two distinct real roots.

x = (−5 ± √1) / 2 = (−5 ± 1) / 2

x₁ = (−5 + 1) / 2 = −2, x₂ = (−5 − 1) / 2 = −3.

Roots: x = −2 and x = −3.

Example 2: x² − 6x + 9 = 0

Coefficients: a = 1, b = −6, c = 9.

Discriminant: Δ = (−6)² − 4×1×9 = 36 − 36 = 0.

Since Δ = 0, there is one repeated real root.

x = (6 ± √0) / 2 = 6 / 2 = 3.

Root: x = 3 (double root).

Example 3: x² + 2x + 5 = 0

Coefficients: a = 1, b = 2, c = 5.

Discriminant: Δ = 2² − 4×1×5 = 4 − 20 = −16.

Since Δ < 0, there are two complex roots.

x = (−2 ± √(−16)) / 2 = (−2 ± 4i) / 2 = −1 ± 2i.

Roots: x = −1 + 2i and x = −1 − 2i.

Example 4: 2x² + 4x − 6 = 0

Coefficients: a = 2, b = 4, c = −6.

Discriminant: Δ = 4² − 4×2×(−6) = 16 + 48 = 64.

Since Δ > 0, two distinct real roots.

x = (−4 ± √64) / 4 = (−4 ± 8) / 4

x₁ = (−4 + 8) / 4 = 1, x₂ = (−4 − 8) / 4 = −3.

Roots: x = 1 and x = −3.

Alternative Methods for Solving Quadratics

While the quadratic formula always works, other methods can be faster in specific cases:

Factoring: If the quadratic factors easily (e.g., x² + 5x + 6 = (x+2)(x+3)), set each factor to zero.
Completing the Square: Rewrite the equation in the form (x + p)² = q.
Graphing: The roots are the x-intercepts of the parabola y = ax² + bx + c.
Square Root Property: For equations of the form x² = k, take the square root of both sides.

Our calculator focuses on the quadratic formula because it is universal — it works for every quadratic equation, including those with complex roots.

Applications of Quadratic Equations

Quadratic equations are used extensively in real life:

Physics: Projectile motion, free fall, and energy problems.
Engineering: Structural design, signal processing, and control systems.
Finance: Calculating profit maximization and break-even points.
Geometry: Finding dimensions of shapes with given area or perimeter.
Computer Graphics: Rendering curves and collision detection.
Optimization: Finding maximum or minimum values of quadratic functions.

Quadratic Formula in Programming

If you are a developer, you may need to solve quadratic equations programmatically. Here is how it looks in several languages:

Python

import cmath

def solve_quadratic(a, b, c):
    if a == 0:
        raise ValueError("a cannot be 0")
    discriminant = b**2 - 4*a*c
    sqrt_disc = cmath.sqrt(discriminant)
    x1 = (-b + sqrt_disc) / (2*a)
    x2 = (-b - sqrt_disc) / (2*a)
    return x1, x2, discriminant

# Example
x1, x2, d = solve_quadratic(1, 5, 6)
print(f"Roots: {x1}, {x2}")  # Roots: -2.0, -3.0
print(f"Discriminant: {d}")   # Discriminant: 1

JavaScript

function solveQuadratic(a, b, c) {
  if (a === 0) throw new Error("a cannot be 0");
  const discriminant = b * b - 4 * a * c;
  if (discriminant >= 0) {
    const sqrtD = Math.sqrt(discriminant);
    return {
      x1: (-b + sqrtD) / (2 * a),
      x2: (-b - sqrtD) / (2 * a),
      discriminant,
      complex: false
    };
  } else {
    const sqrtD = Math.sqrt(-discriminant);
    return {
      real: -b / (2 * a),
      imag: sqrtD / (2 * a),
      discriminant,
      complex: true
    };
  }
}

console.log(solveQuadratic(1, 5, 6)); // { x1: -2, x2: -3, discriminant: 1, complex: false }

C++

#include <iostream>
#include <cmath>
#include <complex>

void solveQuadratic(double a, double b, double c) {
    double discriminant = b * b - 4 * a * c;
    if (discriminant >= 0) {
        double sqrtD = std::sqrt(discriminant);
        std::cout << "x1 = " << (-b + sqrtD) / (2 * a) << std::endl;
        std::cout << "x2 = " << (-b - sqrtD) / (2 * a) << std::endl;
    } else {
        std::complex<double> sqrtD = std::sqrt(std::complex<double>(discriminant, 0));
        std::cout << "x1 = " << (-b + sqrtD) / (2.0 * a) << std::endl;
        std::cout << "x2 = " << (-b - sqrtD) / (2.0 * a) << std::endl;
    }
}

int main() {
    solveQuadratic(1, 5, 6);
    return 0;
}

Historical Background of the Quadratic Formula

The quadratic formula has a rich history. Babylonian mathematicians (c. 2000 BCE) solved quadratic equations using geometric methods. The Greeks, including Euclid and Diophantus, developed algebraic approaches. Indian mathematicians Brahmagupta (c. 628 CE) and Sridhara (c. 870 CE) gave early forms of the quadratic formula. The modern algebraic formula was popularized in Europe during the Renaissance, with contributions from Viète, Descartes, and others. Today, it is one of the most widely known formulas in mathematics.

Common Mistakes to Avoid

Forgetting the ± sign: There are generally two roots, not one.
Incorrect sign for b: The formula uses −b, so if b is negative, −b becomes positive.
Dividing by 2a, not 2: The entire numerator is divided by 2a.
Miscomputing the discriminant: Δ = b² − 4ac (not b² + 4ac).
Setting a = 0: If a = 0, the equation is linear, not quadratic.
Ignoring complex roots: When Δ < 0, the roots are complex (involving i).
Rounding too early: Keep full precision until the final step.

Conclusion

The Quadratic Formula Calculator on lcmcalculator.info is a complete, free, and easy-to-use tool for solving any quadratic equation. With step-by-step solutions, discriminant analysis, and support for both real and complex roots, it is suitable for students, teachers, and professionals alike. Whether you are checking homework, preparing for an exam, or solving a real-world problem, this online quadratic formula calculator will save you time and help you understand the underlying mathematics. Bookmark this page and use it whenever you need to solve a quadratic equation.

Frequently Asked Questions About the Quadratic Formula

Answers to the most common questions about quadratic equations.

What is the quadratic formula? ↓

The quadratic formula is x = (−b ± √(b² − 4ac)) / (2a), used to solve quadratic equations of the form ax² + bx + c = 0 where a ≠ 0.

What is the discriminant? ↓

The discriminant is Δ = b² − 4ac. It determines the nature of the roots: Δ > 0 gives two distinct real roots, Δ = 0 gives one repeated real root, and Δ < 0 gives two complex conjugate roots.

How do you solve x² + 5x + 6 = 0? ↓

Using the quadratic formula with a=1, b=5, c=6: Δ = 25 − 24 = 1. x = (−5 ± 1) / 2, giving x = −2 and x = −3.

Can the quadratic formula calculator handle complex roots? ↓

Yes, when the discriminant is negative, the calculator returns complex conjugate roots in the form a ± bi.

What if a = 0? ↓

If a = 0, the equation is not quadratic — it becomes linear (bx + c = 0). The quadratic formula requires a ≠ 0.

Is this quadratic formula calculator free to use? ↓

Yes, this quadratic formula calculator is completely free with no registration required, no usage limits, and no hidden fees.