Quadratic Equation Solution Calculator

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Quadratic Equation Solver

Enter the coefficients a, b, and c for the quadratic equation in the standard form ax² + bx + c = 0 to find its solutions (roots).

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'); resultDiv.innerHTML = "; // Clear previous results if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients (a, b, and c).'; return; } var output = "; if (a === 0) { // This is a linear equation: bx + c = 0 if (b === 0) { if (c === 0) { output = 'This is the trivial equation 0 = 0. It has infinitely many solutions.'; } else { output = 'This is the equation ' + c + ' = 0, which is false. There are no solutions.'; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; output = 'Since coefficient \'a\' is 0, this is a linear equation.'; output += 'Solution (x): ' + x.toFixed(6) + ''; } } else { // Quadratic equation: ax^2 + bx + c = 0 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 = 'The equation has two distinct real solutions:'; output += 'x₁ = ' + x1.toFixed(6) + ''; output += 'x₂ = ' + x2.toFixed(6) + ''; } else if (discriminant === 0) { var x = -b / (2 * a); output = 'The equation has one real solution (or two identical real solutions):'; output += 'x = ' + x.toFixed(6) + ''; } else { // discriminant < 0 var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); output = 'The equation has two complex conjugate solutions:'; output += 'x₁ = ' + realPart.toFixed(6) + ' + ' + imaginaryPart.toFixed(6) + 'i'; output += 'x₂ = ' + realPart.toFixed(6) + ' – ' + imaginaryPart.toFixed(6) + 'i'; } } resultDiv.innerHTML = output; } .quadratic-calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; padding: 25px; border-radius: 10px; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.1); max-width: 600px; margin: 20px auto; border: 1px solid #eee; } .quadratic-calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .quadratic-calculator-container p { color: #555; line-height: 1.6; margin-bottom: 15px; } .calculator-input-group { margin-bottom: 15px; display: flex; flex-direction: column; } .calculator-input-group label { margin-bottom: 8px; color: #333; font-weight: bold; font-size: 1.1em; } .calculator-input-group input[type="number"] { padding: 12px; border: 1px solid #ccc; border-radius: 6px; font-size: 1em; width: 100%; box-sizing: border-box; transition: border-color 0.3s ease; } .calculator-input-group input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 0 3px rgba(0, 123, 255, 0.25); } .quadratic-calculator-container button { background-color: #007bff; color: white; padding: 12px 25px; border: none; border-radius: 6px; cursor: pointer; font-size: 1.1em; font-weight: bold; width: 100%; transition: background-color 0.3s ease, transform 0.2s ease; margin-top: 10px; } .quadratic-calculator-container button:hover { background-color: #0056b3; transform: translateY(-2px); } .calculator-result { margin-top: 25px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 8px; color: #155724; font-size: 1.1em; line-height: 1.6; } .calculator-result p { margin: 5px 0; } .calculator-result strong { color: #004085; }

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of the second degree. The general form is written as:

ax² + bx + c = 0

where x represents an unknown variable, and a, b, and c are numerical coefficients. A critical condition for an equation to be quadratic is that the coefficient a must not be equal to zero. If a were zero, the ax² term would vanish, and the equation would simplify to a linear equation (bx + c = 0).

The Quadratic Formula

The solutions (also known as roots) for x in a quadratic equation can be found using the quadratic formula. This formula is derived by completing the square and provides a direct method to calculate the values of x that satisfy the equation:

x = [-b ± √(b² - 4ac)] / 2a

The '±' symbol indicates that there are generally two solutions, one using the plus sign and one using the minus sign.

The Discriminant (b² – 4ac)

The term inside the square root, b² - 4ac, is of particular importance and is called the discriminant (often denoted by the Greek letter Delta, Δ). Its value determines the nature of the solutions:

  • If Δ > 0 (Positive Discriminant): There are two distinct real solutions. This means the graph of the quadratic equation (a parabola) intersects the x-axis at two different points.
  • If Δ = 0 (Zero Discriminant): There is exactly one real solution. This is sometimes referred to as a repeated root or two identical real solutions. Graphically, the parabola touches the x-axis at exactly one point (its vertex lies on the x-axis).
  • If Δ < 0 (Negative Discriminant): There are two complex conjugate solutions. In this case, the parabola does not intersect the x-axis; its vertex is either entirely above or entirely below the x-axis. Complex solutions involve the imaginary unit 'i', where i = √(-1).

How to Use This Calculator

  1. Identify Coefficients: Begin by ensuring your quadratic equation is in the standard form ax² + bx + c = 0. Then, identify the numerical values for a, b, and c. Pay close attention to their signs (positive or negative).
  2. Enter Values: Input these numerical values into the corresponding fields: 'Coefficient a', 'Coefficient b', and 'Coefficient c'.
  3. Calculate: Click the "Calculate Solutions" button.
  4. View Results: The calculator will instantly display the solutions for x, clearly indicating whether they are real or complex, and if there are one or two distinct solutions.

Examples

Let's explore a few examples to illustrate the different types of solutions:

Example 1: Two Distinct Real Solutions

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

  • a = 1
  • b = -3
  • c = 2

Using the calculator, you would input 1, -3, and 2. The discriminant would be (-3)² – 4(1)(2) = 9 – 8 = 1 (which is > 0).

Solutions: x₁ = 2, x₂ = 1

Example 2: One Real Solution (Repeated Root)

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

  • a = 1
  • b = 4
  • c = 4

Inputting these values (1, 4, 4) into the calculator. The discriminant would be (4)² – 4(1)(4) = 16 – 16 = 0.

Solution: x = -2

Example 3: Two Complex Conjugate Solutions

Consider the equation: x² + x + 1 = 0

  • a = 1
  • b = 1
  • c = 1

Entering 1, 1, and 1 into the calculator. The discriminant would be (1)² – 4(1)(1) = 1 – 4 = -3 (which is < 0).

Solutions: x₁ = -0.5 + 0.866025i, x₂ = -0.5 – 0.866025i

Example 4: Linear Equation (when 'a' is 0)

Consider the equation: 0x² + 5x - 10 = 0 (which simplifies to 5x - 10 = 0)

  • a = 0
  • b = 5
  • c = -10

Inputting 0, 5, and -10. The calculator will recognize that 'a' is 0 and solve it as a linear equation.

Solution: x = 2

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