Solve Equations Calculator

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Quadratic Equation Solver

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

function calculateQuadraticRoots() { var a = parseFloat(document.getElementById('coefficientA').value); var b = parseFloat(document.getElementById('coefficientB').value); var c = parseFloat(document.getElementById('coefficientC').value); var resultDiv = document.getElementById('quadraticResult'); var output = "; if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients.'; return; } if (a === 0) { // This is a linear equation: bx + c = 0 if (b === 0) { if (c === 0) { output = 'This equation simplifies to 0 = 0, which means there are infinitely many solutions.'; } else { output = 'This equation simplifies to ' + c + ' = 0, which has no solution.'; } } else { var x = -c / b; output = 'This is a linear equation (a = 0). One real root: x = ' + x.toFixed(4) + ''; } } else { // This is a quadratic equation 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); output = 'Two distinct real roots:'; output += 'x₁ = ' + x1.toFixed(4) + ''; output += 'x₂ = ' + x2.toFixed(4) + ''; } else if (discriminant === 0) { var x = -b / (2 * a); output = 'One real root (or two identical real roots):'; output += 'x = ' + x.toFixed(4) + ''; } else { var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); output = 'Two complex conjugate roots:'; output += 'x₁ = ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i'; output += 'x₂ = ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i'; } } resultDiv.innerHTML = output; } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; } .calculator-container p { color: #555; line-height: 1.6; } .form-group { margin-bottom: 15px; } .form-group label { display: block; margin-bottom: 5px; color: #333; font-weight: bold; } .form-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculate-button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; box-sizing: border-box; transition: background-color 0.3s ease; } .calculate-button:hover { background-color: #0056b3; } .calculator-result { margin-top: 20px; padding: 15px; border: 1px solid #e0e0e0; border-radius: 4px; background-color: #e9ecef; min-height: 50px; } .calculator-result p { margin: 5px 0; color: #333; } .calculator-result p.error { color: #dc3545; font-weight: bold; } .calculator-result strong { color: #0056b3; }

Understanding the Quadratic Equation and Its Solutions

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

  • x represents the unknown variable.
  • a, b, and c are coefficients, with a not equal to zero. If a were zero, the equation would become a linear equation (bx + c = 0).

The Quadratic Formula

The most common method to find the solutions (also known as roots) for a quadratic equation is by using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

This formula provides two potential values for x, which are the points where the parabola (the graph of a quadratic equation) intersects the x-axis.

The Discriminant (b² – 4ac)

A crucial part of the quadratic formula is the expression under the square root, (b² - 4ac), which is called the discriminant. The value of the discriminant determines the nature of the roots:

  • If (b² - 4ac) > 0: There are two distinct real roots. This means the parabola intersects the x-axis at two different points.
  • If (b² - 4ac) = 0: There is exactly one real root (sometimes called a repeated root or two identical real roots). The parabola touches the x-axis at exactly one point.
  • If (b² - 4ac) < 0: There are two complex conjugate roots. The parabola does not intersect the x-axis; its vertex is either entirely above or entirely below the x-axis. Complex roots involve the imaginary unit i, where i = √(-1).

How to Use the Quadratic Equation Solver

Our calculator simplifies the process of finding the roots of any quadratic equation. Simply input the coefficients a, b, and c from your equation ax² + bx + c = 0 into the respective fields. The calculator will then apply the quadratic formula and display the roots, indicating whether they are real or complex.

Examples of Quadratic Equation Solutions

Example 1: Two Distinct Real Roots

Consider the equation: x² - 5x + 6 = 0

  • a = 1
  • b = -5
  • c = 6

Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1. Since 1 > 0, there are two distinct real roots.

Using the calculator with these values will yield:

  • x₁ = 3.0000
  • x₂ = 2.0000

Example 2: One Real Root (Repeated)

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

  • a = 1
  • b = -4
  • c = 4

Discriminant = (-4)² - 4(1)(4) = 16 - 16 = 0. Since the discriminant is 0, there is one real root.

Using the calculator with these values will yield:

  • x = 2.0000

Example 3: Two Complex Conjugate Roots

Consider the equation: x² + x + 1 = 0

  • a = 1
  • b = 1
  • c = 1

Discriminant = (1)² - 4(1)(1) = 1 - 4 = -3. Since -3 < 0, there are two complex conjugate roots.

Using the calculator with these values will yield:

  • x₁ = -0.5000 + 0.8660i
  • x₂ = -0.5000 - 0.8660i

Example 4: Linear Equation (when a = 0)

Consider the equation: 0x² + 2x - 4 = 0 (which simplifies to 2x - 4 = 0)

  • a = 0
  • b = 2
  • c = -4

The calculator will recognize this as a linear equation and solve it accordingly.

Using the calculator with these values will yield:

  • x = 2.0000

Example 5: Degenerate Case (when a = 0, b = 0)

Consider the equation: 0x² + 0x + 5 = 0 (which simplifies to 5 = 0)

  • a = 0
  • b = 0
  • c = 5

The calculator will identify that this equation has no solution, as 5 can never equal 0.

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