Quadratic Function Calculator

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Quadratic Function Solver

Results:

Root 1 (x₁):

Root 2 (x₂):

function calculateQuadratic() { var a = parseFloat(document.getElementById("coefficientA").value); var b = parseFloat(document.getElementById("coefficientB").value); var c = parseFloat(document.getElementById("coefficientC").value); var root1Display = document.getElementById("root1Display"); var root2Display = document.getElementById("root2Display"); var rootNature = document.getElementById("rootNature"); root1Display.innerHTML = "Root 1 (x₁):"; root2Display.innerHTML = "Root 2 (x₂):"; rootNature.innerHTML = ""; if (isNaN(a) || isNaN(b) || isNaN(c)) { rootNature.innerHTML = "Please enter valid numbers for all coefficients."; return; } if (a === 0) { rootNature.innerHTML = "If 'a' is 0, this is a linear equation, not a quadratic equation."; if (b !== 0) { var x = -c / b; root1Display.innerHTML = "Solution (x): " + x.toFixed(4); root2Display.innerHTML = ""; } else if (c === 0) { root1Display.innerHTML = "Infinite solutions (0 = 0)."; root2Display.innerHTML = ""; } else { root1Display.innerHTML = "No solution (e.g., 5 = 0)."; root2Display.innerHTML = ""; } 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); root1Display.innerHTML = "Root 1 (x₁): " + x1.toFixed(4); root2Display.innerHTML = "Root 2 (x₂): " + x2.toFixed(4); rootNature.innerHTML = "Nature of Roots: Two distinct real roots."; } else if (discriminant === 0) { var x = -b / (2 * a); root1Display.innerHTML = "Root (x): " + x.toFixed(4); root2Display.innerHTML = "Root (x): " + x.toFixed(4) + " (repeated)"; rootNature.innerHTML = "Nature of Roots: One real (repeated) root."; } else { // discriminant < 0 var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); root1Display.innerHTML = "Root 1 (x₁): " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i"; root2Display.innerHTML = "Root 2 (x₂): " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; rootNature.innerHTML = "Nature of Roots: Two complex conjugate roots."; } }

Understanding the Quadratic Function and Its Roots

A quadratic function is a polynomial function of the form f(x) = ax² + bx + c, where 'a', 'b', and 'c' are coefficients, and 'a' is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that can open upwards or downwards. The 'roots' or 'zeros' of a quadratic function are the x-values where the parabola intersects the x-axis, meaning f(x) = 0.

The Quadratic Formula

To find the roots of a quadratic equation ax² + bx + c = 0, we use the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

This formula provides a direct way to calculate the values of 'x' that satisfy the equation, given the coefficients 'a', 'b', and 'c'.

The Discriminant: Unveiling the Nature of Roots

A crucial part of the quadratic formula is the expression under the square root: b² - 4ac. This term is called the discriminant, often denoted by the Greek letter Delta (Δ). The value of the discriminant tells us about the nature of the roots without actually calculating them:

  • 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 (also called a repeated root or a double root). In this case, the parabola touches the x-axis at exactly one point, which is 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 This Calculator

Our Quadratic Function Solver simplifies the process of finding the roots of any quadratic equation. Simply input the coefficients 'a', 'b', and 'c' from your equation ax² + bx + c = 0 into the respective fields. Click the "Calculate Roots" button, and the calculator will instantly display the roots (x₁ and x₂) and describe their nature (real, repeated, or complex).

Examples of Quadratic Equations and Their Roots

Example 1: Two Distinct Real Roots

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

  • a = 1
  • b = -3
  • c = 2

Discriminant (Δ) = (-3)² – 4(1)(2) = 9 – 8 = 1. Since Δ > 0, there are two distinct real roots.

Using the calculator with these values will yield:

  • x₁ = 2.0000
  • x₂ = 1.0000

Example 2: One Real (Repeated) Root

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

  • a = 1
  • b = -4
  • c = 4

Discriminant (Δ) = (-4)² – 4(1)(4) = 16 – 16 = 0. Since Δ = 0, there is one real repeated root.

Using the calculator with these values will yield:

  • x = 2.0000 (repeated)

Example 3: Two Complex Conjugate Roots

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

  • a = 1
  • b = 2
  • c = 5

Discriminant (Δ) = (2)² – 4(1)(5) = 4 – 20 = -16. Since Δ < 0, there are two complex conjugate roots.

Using the calculator with these values will yield:

  • x₁ = -1.0000 + 2.0000i
  • x₂ = -1.0000 – 2.0000i

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