BeastCalculator

Quadratic Equation Solver

Solve quadratic equations instantly with step-by-step solutions. Enter your coefficients and get detailed analysis including discriminant calculation and root classification.

Enter Quadratic Equation Coefficients
Enter the coefficients for your quadratic equation in the form ax² + bx + c = 0

Coefficient of x²

Coefficient of x

Constant term

Your equation:

1x² +0x +0 = 0

What is a Quadratic Equation?

A quadratic equation is a second-degree polynomial equation in a single variable x, with the general form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. These equations are fundamental in algebra and appear frequently in physics, engineering, economics, and many other fields.

The solutions to quadratic equations, called roots, represent the x-values where the parabola intersects the x-axis. Understanding how to solve these equations is crucial for advanced mathematics and real-world problem-solving.

How to Use the Quadratic Equation Solver

Step 1: Enter Coefficients

Input the values for coefficients a, b, and c from your quadratic equation ax² + bx + c = 0. The coefficient 'a' cannot be zero.

Step 2: Calculate

Click the "Solve Equation" button to instantly calculate the discriminant, determine root types, and find the solutions.

Step 3: Analyze Results

Review the step-by-step solution, discriminant analysis, and root classification to understand the complete solution process.

Step 4: Share or Reset

Use the shareable URL to save your results or click reset to solve another equation with different coefficients.

Quadratic Formula and Methodology

The Quadratic Formula

x = (-b ± √(b² - 4ac)) / (2a)

This formula provides the exact solutions to any quadratic equation ax² + bx + c = 0 where a ≠ 0.

Discriminant Analysis

The discriminant Δ = b² - 4ac determines the nature of the roots:

  • • Δ > 0: Two distinct real roots
  • • Δ = 0: One repeated real root (perfect square)
  • • Δ < 0: Two complex conjugate roots

Solution Process

  1. 1. Identify coefficients a, b, and c from the standard form
  2. 2. Calculate the discriminant: Δ = b² - 4ac
  3. 3. Determine root type based on discriminant value
  4. 4. Apply the quadratic formula to find solutions
  5. 5. Simplify and present the final answers

Mathematical References

Our quadratic equation solver is based on established mathematical principles and formulas:

  • • Quadratic Formula: Derived from completing the square method, first documented by ancient Babylonian mathematicians
  • • Discriminant Theory: Developed by mathematicians including Vieta and Descartes
  • • Complex Number Theory: Extended by Euler and Gauss for negative discriminant cases
  • • Algebraic Methods: Based on fundamental theorem of algebra and polynomial theory

Frequently Asked Questions

What is a quadratic equation?

A quadratic equation is a polynomial equation of degree 2, typically written in the form ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0. The highest power of the variable is 2.

How does the quadratic formula work?

The quadratic formula x = (-b ± √(b² - 4ac)) / (2a) provides the solutions to any quadratic equation. The discriminant (b² - 4ac) determines the nature of the roots: positive for two real roots, zero for one repeated root, and negative for complex roots.

What is the discriminant in quadratic equations?

The discriminant is the expression b² - 4ac under the square root in the quadratic formula. It determines the nature of the roots: if positive, there are two distinct real roots; if zero, there's one repeated real root; if negative, there are two complex conjugate roots.

Can this calculator solve all types of quadratic equations?

Yes, our calculator can solve all quadratic equations including those with real distinct roots, repeated real roots, and complex roots. It provides step-by-step solutions and explains the nature of each solution type.

Authoritative References

For further reading and verification of quadratic equation methods, consult these authoritative sources: