Quadratic Function Equation Calculator

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Quadratic Function Equation Calculator

Calculation Results

function calculateQuadratic() { var a = parseFloat(document.getElementById('coeffA').value); var b = parseFloat(document.getElementById('coeffB').value); var c = parseFloat(document.getElementById('coeffC').value); var resultArea = document.getElementById('quad-result-area'); if (isNaN(a) || isNaN(b) || isNaN(c)) { alert("Please enter valid numerical values for a, b, and c."); return; } if (a === 0) { alert("The value of 'a' cannot be zero in a quadratic equation. This would be a linear equation."); return; } var discriminant = (b * b) – (4 * a * c); var vertexX = -b / (2 * a); var vertexY = (a * Math.pow(vertexX, 2)) + (b * vertexX) + c; var rootsText = ""; if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); rootsText = "Two Real Roots: x₁ = " + x1.toFixed(4) + ", x₂ = " + x2.toFixed(4); } else if (discriminant === 0) { var x = -b / (2 * a); rootsText = "One Real Root (Double Root): x = " + x.toFixed(4); } else { var realPart = (-b / (2 * a)).toFixed(4); var imagPart = (Math.sqrt(-discriminant) / (2 * a)).toFixed(4); rootsText = "Complex Roots: x = " + realPart + " ± " + Math.abs(imagPart) + "i"; } document.getElementById('equation-display').innerHTML = "Equation: " + a + "x² + (" + b + ")x + (" + c + ") = 0"; document.getElementById('discriminant-val').innerHTML = "Discriminant (Δ): " + discriminant.toFixed(4); document.getElementById('roots-val').innerHTML = "Roots: " + rootsText; document.getElementById('vertex-val').innerHTML = "Vertex (h, k): (" + vertexX.toFixed(4) + ", " + vertexY.toFixed(4) + ")"; document.getElementById('symmetry-val').innerHTML = "Axis of Symmetry: x = " + vertexX.toFixed(4); resultArea.style.display = "block"; }

Understanding Quadratic Functions

A quadratic function is a second-degree polynomial function. The general form of a quadratic equation is expressed as f(x) = ax² + bx + c, where a, b, and c are constants (coefficients) and a ≠ 0.

Key Components of a Quadratic Equation

  • The Quadratic Term: ax², where a determines whether the parabola opens upward (positive) or downward (negative).
  • The Linear Term: bx, which affects the position of the axis of symmetry.
  • The Constant Term: c, which represents the y-intercept of the function's graph.

Solving for Roots: The Quadratic Formula

To find the values of x where the function equals zero (the x-intercepts), we use the quadratic formula:

x = [-b ± sqrt(b² – 4ac)] / (2a)

The Role of the Discriminant

The expression inside the square root, b² – 4ac, is known as the discriminant (Δ). It provides vital information about the nature of the roots:

  • Δ > 0: Two distinct real roots (the graph crosses the x-axis twice).
  • Δ = 0: One real root (the graph touches the x-axis at exactly one point).
  • Δ < 0: Two complex (imaginary) roots (the graph does not touch the x-axis).

The Vertex and Axis of Symmetry

Every quadratic function produces a "U-shaped" graph called a parabola. The lowest or highest point on the parabola is the vertex. The x-coordinate of the vertex is found using the formula h = -b / (2a). The axis of symmetry is the vertical line x = h that passes through the vertex, dividing the parabola into two mirrored halves.

Practical Example

Suppose you have the equation: 2x² – 4x – 6 = 0.

  • Identify coefficients: a = 2, b = -4, c = -6.
  • Calculate Discriminant: (-4)² – 4(2)(-6) = 16 + 48 = 64.
  • Find Roots: x = [4 ± sqrt(64)] / 4. This gives x = 3 and x = -1.
  • Find Vertex: x = -(-4) / (2 * 2) = 1. Plugging 1 back into the equation gives y = 2(1)² – 4(1) – 6 = -8. The vertex is (1, -8).

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