Mathematic Calculator

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Quadratic Equation Solver

Enter the coefficients (a, b, c) of your quadratic equation in the standard form ax² + bx + c = 0 to find its 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('result'); if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = 'Please enter valid numbers for all coefficients.'; return; } if (a === 0) { resultDiv.innerHTML = 'Coefficient \'a\' cannot be zero for a quadratic equation. This would be a linear equation.'; return; } var discriminant = b * b – 4 * a * c; var x1, x2; if (discriminant > 0) { x1 = (-b + Math.sqrt(discriminant)) / (2 * a); 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) { x1 = -b / (2 * a); resultDiv.innerHTML = 'The equation has one real root (or two identical real roots):' + 'Root (x): ' + x1.toFixed(4) + ''; } else { var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultDiv.innerHTML = 'The equation has two complex conjugate roots:' + 'Root 1 (x₁): ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i' + 'Root 2 (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-container p { color: #555; line-height: 1.6; } .form-group { margin-bottom: 15px; } .form-group label { display: block; margin-bottom: 5px; font-weight: bold; color: #444; } .form-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-container button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; display: block; margin-top: 20px; } .calculator-container button:hover { background-color: #0056b3; } .calculator-result { margin-top: 25px; padding: 15px; border: 1px solid #e0e0e0; border-radius: 4px; background-color: #e9ecef; color: #333; } .calculator-result p { margin: 5px 0; font-size: 1.1em; } .calculator-result p strong { color: #0056b3; } .calculator-result .error { color: #dc3545; font-weight: bold; }

Understanding Quadratic Equations and Their Roots

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:

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

The solutions to a quadratic equation are called its "roots" or "zeros." These are the values of 'x' that make the equation true. Graphically, 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 to find the roots of a quadratic equation is by using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

This formula allows you to directly calculate the values of 'x' given the coefficients a, b, and c.

The Discriminant (b² – 4ac)

A crucial part of the quadratic formula is the expression under the square root, (b² - 4ac). This is known as the discriminant, often denoted by the Greek letter Delta (Δ). The value of the discriminant tells us about the nature of the roots:

  • 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 the imaginary unit 'i' (where i² = -1).

How to Use the Calculator

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

  • Equation: x² – 5x + 6 = 0
    • a = 1, b = -5, c = 6
    • Discriminant = (-5)² – 4(1)(6) = 25 – 24 = 1 (Positive)
    • Roots: x₁ = 3, x₂ = 2 (Two distinct real roots)
  • Equation: x² – 4x + 4 = 0
    • a = 1, b = -4, c = 4
    • Discriminant = (-4)² – 4(1)(4) = 16 – 16 = 0 (Zero)
    • Root: x = 2 (One real root)
  • Equation: x² + x + 1 = 0
    • a = 1, b = 1, c = 1
    • Discriminant = (1)² – 4(1)(1) = 1 – 4 = -3 (Negative)
    • Roots: x₁ = -0.5 + 0.8660i, x₂ = -0.5 – 0.8660i (Two complex conjugate roots)

This tool is invaluable for students, engineers, and anyone needing to quickly solve quadratic equations without manual calculation.

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