Math Wolfram Calculator

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Quadratic Equation Solver

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.'; return; } if (a === 0) { if (b === 0) { if (c === 0) { resultDiv.innerHTML = 'The equation is 0 = 0, which is true for all x (infinite 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 root is x = ' + x.toFixed(4) + '.'; } return; } var discriminant = b * b – 4 * a * c; var root1, root2; if (discriminant > 0) { root1 = (-b + Math.sqrt(discriminant)) / (2 * a); root2 = (-b – Math.sqrt(discriminant)) / (2 * a); resultDiv.innerHTML = 'The equation has two distinct real roots:' + 'Root 1 (x₁): ' + root1.toFixed(4) + " + 'Root 2 (x₂): ' + root2.toFixed(4) + "; } else if (discriminant === 0) { root1 = -b / (2 * a); resultDiv.innerHTML = 'The equation has one real root (or two identical real roots):' + 'Root (x): ' + root1.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:' + '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: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calculator-form .form-group { margin-bottom: 15px; } .calculator-form label { display: block; margin-bottom: 5px; color: #555; font-weight: bold; } .calculator-form input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; font-size: 16px; } .calculator-form input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 5px rgba(0, 123, 255, 0.25); } .calculate-button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 18px; cursor: pointer; transition: background-color 0.3s ease; } .calculate-button:hover { background-color: #0056b3; } .calculator-result { margin-top: 25px; padding: 15px; background-color: #e9ecef; border: 1px solid #ced4da; border-radius: 4px; min-height: 50px; color: #333; } .calculator-result p { margin: 5px 0; font-size: 1.1em; line-height: 1.5; } .calculator-result p.error { color: #dc3545; font-weight: bold; }

Understanding and Solving Quadratic Equations

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 'x' represents an unknown, and 'a', 'b', and 'c' are coefficients, with 'a' not equal to zero. The solutions to a quadratic equation are called its roots, and they represent the x-intercepts of the parabola defined by the equation when graphed.

The Quadratic Formula

The most common and reliable method for finding the roots of any quadratic equation is the quadratic formula:

x = [-b ± sqrt(b² - 4ac)] / 2a

This formula allows you to directly calculate the values of 'x' that satisfy the equation, 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, 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: The equation has two distinct real roots. This means the parabola intersects the x-axis at two different points.
  • If Δ = 0: The equation has exactly one real root (sometimes called a repeated or double root). The parabola touches the x-axis at exactly one point.
  • If Δ < 0: The equation has two complex conjugate roots. In this case, the parabola does not intersect the x-axis at all.

How to Use the Quadratic Equation Solver

Our "Wolfram-like" quadratic equation solver simplifies the process of finding roots. Simply input the coefficients 'a', 'b', and 'c' from your quadratic equation into the respective fields:

  1. Coefficient 'a': Enter the number multiplying x². If there's no number, it's usually 1. (e.g., for x² – 3x + 2 = 0, 'a' is 1)
  2. Coefficient 'b': Enter the number multiplying x. (e.g., for x² – 3x + 2 = 0, 'b' is -3)
  3. Coefficient 'c': Enter the constant term. (e.g., for x² – 3x + 2 = 0, 'c' is 2)

Click the "Calculate Roots" button, and the calculator will instantly display the roots of your equation, along with their nature (real or complex).

Examples of Quadratic Equations and Their Solutions

  • Example 1: Two Distinct Real Roots

    Equation: x² - 5x + 6 = 0

    Here, a=1, b=-5, c=6.

    Discriminant = (-5)² – 4(1)(6) = 25 – 24 = 1 (Δ > 0)

    Roots: x₁ = [5 + sqrt(1)] / 2 = 3, x₂ = [5 – sqrt(1)] / 2 = 2

    Using the calculator: Input a=1, b=-5, c=6. Output: Root 1: 3.0000, Root 2: 2.0000

  • Example 2: One Real Root (Repeated)

    Equation: x² - 4x + 4 = 0

    Here, a=1, b=-4, c=4.

    Discriminant = (-4)² – 4(1)(4) = 16 – 16 = 0 (Δ = 0)

    Root: x = [4 ± sqrt(0)] / 2 = 2

    Using the calculator: Input a=1, b=-4, c=4. Output: Root: 2.0000

  • Example 3: Two Complex Conjugate Roots

    Equation: x² + 2x + 5 = 0

    Here, a=1, b=2, c=5.

    Discriminant = (2)² – 4(1)(5) = 4 – 20 = -16 (Δ < 0)

    Roots: x₁ = [-2 + sqrt(-16)] / 2 = [-2 + 4i] / 2 = -1 + 2i

    x₂ = [-2 – sqrt(-16)] / 2 = [-2 – 4i] / 2 = -1 – 2i

    Using the calculator: Input a=1, b=2, c=5. Output: Root 1: -1.0000 + 2.0000i, Root 2: -1.0000 – 2.0000i

  • Example 4: Linear Equation (when a=0)

    Equation: 0x² + 3x - 6 = 0 (or simply 3x - 6 = 0)

    Here, a=0, b=3, c=-6.

    Using the calculator: Input a=0, b=3, c=-6. Output: This is a linear equation (a=0). The root is x = 2.0000.

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

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