Welcome to our Quadratic Equation Calculator! This tool helps you solve quadratic equations of the form \(ax^2 + bx + c = 0\). It provides:

Both real and complex solutions
A visual graph of the parabola
Step-by-step solution explanations

How to Use:

Enter the coefficients for your quadratic equation:
- '\(a\)' - coefficient of \(x^2\)
- '\(b\)' - coefficient of \(x\)
- '\(c\)' - constant term
Click "Solve Equation" or press Enter
The calculator will show:
- The solutions \((x_1\)) and \((x_2\))
- A graph showing the parabola and its roots
- Detailed steps of the solution process

\(ax^2 + bx + c = 0\)

Enter a:

Enter b:

Enter c:

First Solution \((x_1\))

Second Solution \((x_2\))

Step-by-Step Solution

About Quadratic Equations

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree 2, written in the form \(ax^2 + bx + c = 0\), where \(a \neq 0\). The solutions to this equation are called roots or zeros, and they represent the x-coordinates where the parabola crosses the x-axis.

Understanding the Graph

The graph of a quadratic equation is called a parabola. When a > 0, it opens upward and has a minimum point. When a < 0, it opens downward and has a maximum point. The roots shown on the graph are the points where the parabola intersects the x-axis.

Types of Solutions

A quadratic equation can have:

Two distinct real solutions (parabola crosses x-axis at two points)
One repeated real solution (parabola touches x-axis at one point)
Two complex/imaginary solutions (parabola doesn't cross x-axis)

Solving the Quadratic Equation Programmatically

This section demonstrates how to solve quadratic equations programmatically in Python, R, Java, and JavaScript. Each implementation highlights the process step-by-step.

Python Implementation

Python provides a straightforward way to solve quadratic equations using the math library. Here's how you can implement it:

Python Code

import math

# Coefficients
a = 1
b = -3
c = 2

# Calculate discriminant
discriminant = b**2 - 4*a*c

# Compute roots
if discriminant > 0:
    root1 = (-b + math.sqrt(discriminant)) / (2 * a)
    root2 = (-b - math.sqrt(discriminant)) / (2 * a)
    print(f"Two real roots: {root1}, {root2}")
elif discriminant == 0:
    root = -b / (2 * a)
    print(f"One real root: {root}")
else:
    real_part = -b / (2 * a)
    imag_part = math.sqrt(-discriminant) / (2 * a)
    print(f"Two complex roots: {real_part}+{imag_part}j, {real_part}-{imag_part}j")

Example Output: Two real roots: 2.0, 1.0

R Implementation

In R, you can use built-in functions like sqrt to calculate roots. Here's an example implementation:

R Code

# Coefficients
a <- 1
b <- 2
c <- 5

# Calculate discriminant
discriminant <- b^2 - 4*a*c

# Compute roots
if (discriminant > 0) {
  root1 <- (-b + sqrt(discriminant)) / (2 * a)
  root2 <- (-b - sqrt(discriminant)) / (2 * a)
  cat("Two real roots:", root1, ",", root2, "\n")
} else if (discriminant == 0) {
  root <- -b / (2 * a)
  cat("One real root:", root, "\n")
} else {
  real_part <- -b / (2 * a)
  imag_part <- sqrt(-discriminant) / (2 * a)
  cat("Two complex roots:", real_part, "+", imag_part, "i,",
      real_part, "-", imag_part, "i\n")
}

Example Output: Two complex roots: -1 + 2i, -1 - 2i

Java Implementation

Java requires explicit handling of calculations using the Math class. Here's an example:

Java Code

public class QuadraticEquation {
    public static void main(String[] args) {
        double a = 1, b = -3, c = 2;
        double discriminant = b * b - 4 * a * c;

        if (discriminant > 0) {
            double root1 = (-b + Math.sqrt(discriminant)) / (2 * a);
            double root2 = (-b - Math.sqrt(discriminant)) / (2 * a);
            System.out.println("Two real roots: " + root1 + ", " + root2);
        } else if (discriminant == 0) {
            double root = -b / (2 * a);
            System.out.println("One real root: " + root);
        } else {
            double realPart = -b / (2 * a);
            double imagPart = Math.sqrt(-discriminant) / (2 * a);
            System.out.println("Two complex roots: " + realPart + "+" + imagPart + "i, " +
                               realPart + "-" + imagPart + "i");
        }
    }
}

Example Output: Two real roots: 2.0, 1.0

JavaScript Implementation

In JavaScript, you can implement the solution using basic math operations. Here's an example:

JavaScript Code

const a = 1, b = -3, c = 2;
const discriminant = b ** 2 - 4 * a * c;

if (discriminant > 0) {
    const root1 = (-b + Math.sqrt(discriminant)) / (2 * a);
    const root2 = (-b - Math.sqrt(discriminant)) / (2 * a);
    console.log(`Two real roots: ${root1}, ${root2}`);
} else if (discriminant === 0) {
    const root = -b / (2 * a);
    console.log(`One real root: ${root}`);
} else {
    const realPart = (-b / (2 * a)).toFixed(3);
    const imagPart = (Math.sqrt(-discriminant) / (2 * a)).toFixed(3);
    console.log(`Two complex roots: ${realPart}+${imagPart}i, ${realPart}-${imagPart}i`);
}

Example Output: Two real roots: 2, 1

Attribution and Citation

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Quadratic Formula Calculator