Math Equation Calculator

by

Quadratic Equation Solver

Enter the coefficients for the quadratic equation in the form: ax² + bx + c = 0







Results:

function calculateQuadraticRoots() { var a = parseFloat(document.getElementById('coeffA').value); var b = parseFloat(document.getElementById('coeffB').value); var c = parseFloat(document.getElementById('coeffC').value); var resultDiv = document.getElementById('quadraticResult'); resultDiv.innerHTML = "; // Clear previous results if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients.'; return; } if (a === 0) { // Not a quadratic equation, it's a linear equation: bx + c = 0 if (b === 0) { if (c === 0) { resultDiv.innerHTML = 'This is the equation 0 = 0, which has infinitely many solutions.'; } else { resultDiv.innerHTML = 'This is the equation ' + c + ' = 0, which has no solution.'; } } else { var x = -c / b; resultDiv.innerHTML = 'This is a linear equation (a=0). The single root is: x = ' + x.toFixed(4) + ''; } return; } var discriminant = b * b – 4 * a * c; if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); resultDiv.innerHTML = 'The equation has two distinct real roots:' + 'x₁ = ' + x1.toFixed(4) + '' + 'x₂ = ' + x2.toFixed(4) + ''; } else if (discriminant === 0) { var x = -b / (2 * a); resultDiv.innerHTML = 'The equation has one real root (or two equal real roots):' + 'x = ' + x.toFixed(4) + ''; } else { // discriminant < 0 var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultDiv.innerHTML = 'The equation has two complex conjugate roots:' + 'x₁ = ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i' + 'x₂ = ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i'; } } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; } .calculator-inputs label { display: inline-block; width: 150px; margin-bottom: 10px; font-weight: bold; } .calculator-inputs input[type="number"] { width: calc(100% – 160px); padding: 8px; margin-bottom: 10px; border: 1px solid #ccc; border-radius: 4px; } .calculator-inputs button { display: block; width: 100%; padding: 10px 15px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 16px; cursor: pointer; margin-top: 15px; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-results { margin-top: 20px; padding-top: 15px; border-top: 1px solid #eee; } .calculator-results h3 { color: #333; margin-bottom: 10px; } .calculator-results p { margin-bottom: 5px; line-height: 1.5; } .calculator-results strong { color: #007bff; }

Understanding the Quadratic Equation and Its Solutions

A quadratic equation is a fundamental concept in algebra, representing a polynomial equation of the second degree. It is typically written in the standard form:

ax² + bx + c = 0

Where:

  • a, b, and c are coefficients, with a not equal to zero.
  • x represents the unknown variable.

The solutions to a quadratic equation are also known as its roots. These roots are the values of x that satisfy the equation, meaning when substituted into the equation, they make the statement true. Geometrically, the roots represent the x-intercepts of the parabola defined by the quadratic function y = ax² + bx + c.

The Quadratic Formula

The most common method for finding the roots of a quadratic equation is using the quadratic formula:

x = [-b ± sqrt(b² - 4ac)] / (2a)

This formula directly provides the values of x based on the coefficients a, b, and c.

The Discriminant (Δ)

A crucial part of the quadratic formula is the expression under the square root: b² - 4ac. This is called the discriminant, often denoted by the Greek letter delta (Δ). The value of the discriminant tells us about the nature of the roots without actually calculating them:

  • If Δ > 0 (Discriminant is positive): The equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
  • If Δ = 0 (Discriminant is zero): The equation has exactly one real root (sometimes called a repeated or double root). This means the parabola touches the x-axis at exactly one point (its vertex).
  • If Δ < 0 (Discriminant is negative): The equation has two complex conjugate roots. This means the parabola does not intersect the x-axis at all. The roots involve imaginary numbers.

How to Use the Quadratic Equation Solver

Our Quadratic Equation Solver simplifies the process of finding these roots. To use it:

  1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember to include their signs (positive or negative).
  2. Enter Values: Input these numerical values into the respective fields: 'Coefficient 'a", 'Coefficient 'b", and 'Constant 'c".
  3. Calculate: Click the "Calculate Roots" button.
  4. View Results: The calculator will instantly display the roots of your equation, indicating whether they are real, equal, or complex.

Examples:

Let's look at some practical examples:

Example 1: Two Distinct Real Roots

Consider the equation: x² - 3x + 2 = 0

  • a = 1
  • b = -3
  • c = 2

Using the calculator with these values will yield:

x₁ = 2.0000

x₂ = 1.0000

Here, the discriminant is (-3)² - 4(1)(2) = 9 - 8 = 1, which is positive, indicating two distinct real roots.

Example 2: One Real Root (Repeated)

Consider the equation: x² - 4x + 4 = 0

  • a = 1
  • b = -4
  • c = 4

The calculator will show:

x = 2.0000

In this case, the discriminant is (-4)² - 4(1)(4) = 16 - 16 = 0, indicating one real root.

Example 3: Two Complex Conjugate Roots

Consider the equation: x² + 2x + 5 = 0

  • a = 1
  • b = 2
  • c = 5

The calculator will output:

x₁ = -1.0000 + 2.0000i

x₂ = -1.0000 – 2.0000i

Here, the discriminant is (2)² - 4(1)(5) = 4 - 20 = -16, which is negative, resulting in complex roots.

Example 4: Linear Equation (a = 0)

Consider the equation: 0x² + 5x - 10 = 0 (or simply 5x - 10 = 0)

  • a = 0
  • b = 5
  • c = -10

The calculator will identify this as a linear equation and provide:

x = 2.0000

This calculator is a powerful tool for students, engineers, and anyone needing to quickly solve quadratic equations without manual calculation errors.

Leave a Comment