Algebraic Equation Calculator

Quadratic Equation Solver

Use this calculator to find the roots (solutions) of a quadratic equation in the standard form: ax² + bx + c = 0.

function calculateQuadratic() { 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) { if (b === 0) { if (c === 0) { resultDiv.innerHTML = 'The equation is 0 = 0, which means there are infinitely many solutions.'; } else { resultDiv.innerHTML = 'The equation is ' + c + ' = 0, which has no solution.'; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; resultDiv.innerHTML = 'This is a linear equation (a=0). The solution 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:' + 'Root 1 (x₁): ' + x1.toFixed(4) + '' + 'Root 2 (x₂): ' + x2.toFixed(4) + ''; } else if (discriminant === 0) { var x = -b / (2 * a); resultDiv.innerHTML = 'The equation has one real root (or two identical real roots):' + 'Root (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 roots:' + 'Root 1 (x₁): ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i' + 'Root 2 (x₂): ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i'; } } .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: 500px; margin: 30px auto; border: 1px solid #e0e0e0; } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .calculator-container p { color: #555; margin-bottom: 15px; line-height: 1.6; } .calc-input-group { margin-bottom: 15px; } .calc-input-group label { display: block; margin-bottom: 7px; color: #444; font-weight: bold; } .calc-input-group input[type="number"] { width: calc(100% – 20px); padding: 10px; border: 1px solid #ccc; border-radius: 5px; font-size: 1em; box-sizing: border-box; } .calc-input-group input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 5px rgba(0, 123, 255, 0.3); } .calculate-button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 5px; font-size: 1.1em; cursor: pointer; transition: background-color 0.3s ease, transform 0.2s ease; margin-top: 20px; } .calculate-button:hover { background-color: #0056b3; transform: translateY(-2px); } .calculate-button:active { background-color: #004085; transform: translateY(0); } .calc-result-area { margin-top: 25px; padding: 15px; background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 8px; min-height: 50px; color: #155724; font-size: 1.1em; line-height: 1.6; } .calc-result-area p { margin: 5px 0; color: #155724; } .calc-result-area strong { color: #004d00; } .calc-result-area .error { color: #721c24; background-color: #f8d7da; border-color: #f5c6cb; padding: 10px; border-radius: 5px; }

Understanding Quadratic Equations and Their 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 written as:

ax² + bx + c = 0

Where:

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

The solutions to a quadratic equation are also known as its "roots" or "zeros." These are the values of x that make the equation true. A quadratic equation can have two distinct real roots, one real root (which is a repeated root), or two complex conjugate roots.

The Quadratic Formula

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

x = [-b ± sqrt(b² - 4ac)] / 2a

This formula directly provides the values of x once the coefficients a, b, and c are known. The term inside the square root, (b² - 4ac), is called the discriminant, often denoted by Δ (Delta). The value of the discriminant determines the nature of the roots:

• If Δ > 0 (positive discriminant): There are two distinct real roots.
• If Δ = 0 (zero discriminant): There is exactly one real root (a repeated root).
• If Δ < 0 (negative discriminant): There are two complex conjugate roots.

How to Use the Quadratic Equation Solver

Our calculator simplifies the process of finding these roots. Here's how to use it:

1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember that a is the coefficient of x², b is the coefficient of x, and c is the constant term.
2. Input Values: Enter these numerical values into the corresponding input fields: "Coefficient 'a' (for x²)", "Coefficient 'b' (for x)", 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 or complex.

Examples of Quadratic Equations

Let's look at a few examples to illustrate different types of solutions:

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:

Root 1 (x₁): 2.0000
Root 2 (x₂): 1.0000

This is because (x-1)(x-2) = x² – 3x + 2.

Example 2: One Real Root (Repeated)

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

• a = 1
• b = -4
• c = 4

Using the calculator with these values will yield:

Root (x): 2.0000

This is because (x-2)² = x² – 4x + 4.

Example 3: Two Complex Roots

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

• a = 1
• b = 2
• c = 5

Using the calculator with these values will yield:

Root 1 (x₁): -1.0000 + 2.0000i
Root 2 (x₂): -1.0000 – 2.0000i

Here, the discriminant (b² – 4ac) = (2)² – 4(1)(5) = 4 – 20 = -16, which is negative, indicating 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

Using the calculator with these values will yield:

This is a linear equation (a=0). The solution is: x = 2.0000

This is because 5x = 10, so x = 2.

This quadratic equation solver is a handy tool for students, engineers, and anyone needing to quickly find the solutions to quadratic equations without manual calculation.