Polynomial Equation Calculator

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Polynomial Equation Solver (Quadratic)

Results:

Enter coefficients and click 'Calculate Roots'.

function calculatePolynomialRoots() { var a = parseFloat(document.getElementById('coefficientA').value); var b = parseFloat(document.getElementById('coefficientB').value); var c = parseFloat(document.getElementById('coefficientC').value); var resultOutput = document.getElementById('resultOutput'); if (isNaN(a) || isNaN(b) || isNaN(c)) { resultOutput.innerHTML = "Please enter valid numbers for all coefficients."; return; } var outputHTML = ""; if (a === 0) { // This is a linear equation: bx + c = 0 if (b === 0) { if (c === 0) { outputHTML = "The equation is 0 = 0, which is true for all x (infinite solutions)."; } else { outputHTML = "The equation is " + c + " = 0, which is false (no solution)."; } } else { var x = -c / b; outputHTML = "This is a linear equation: " + b + "x + " + c + " = 0.The single real root is: x = " + x.toFixed(4); } } else { // This is a quadratic equation: ax^2 + bx + c = 0 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); outputHTML = "The equation has two distinct real roots:"; outputHTML += "x₁ = " + x1.toFixed(4) + ""; outputHTML += "x₂ = " + x2.toFixed(4); } else if (discriminant === 0) { var x = -b / (2 * a); outputHTML = "The equation has one real (repeated) root:"; outputHTML += "x = " + x.toFixed(4); } else { // discriminant < 0, complex roots var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); outputHTML = "The equation has two complex conjugate roots:"; outputHTML += "x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i"; outputHTML += "x₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } } resultOutput.innerHTML = outputHTML; }

Understanding Polynomial Equations and Their Solutions

A polynomial equation is an equation consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. The general form of a polynomial equation is:

anxn + an-1xn-1 + … + a1x + a0 = 0

where 'x' is the variable, 'n' is a non-negative integer representing the degree of the polynomial, and an, an-1, …, a0 are the coefficients. The highest exponent 'n' determines the degree of the polynomial.

Focusing on Quadratic Equations

Our calculator specifically addresses quadratic equations, which are polynomial equations of the second degree. They take the standard form:

ax² + bx + c = 0

Here, 'a', 'b', and 'c' are coefficients, where 'a' cannot be zero (if 'a' were zero, it would become a linear equation). The solutions to a polynomial equation are also known as its 'roots' or 'zeros' – these are the values of 'x' that make the equation true.

The Quadratic Formula

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

x = [-b ± √(b² – 4ac)] / 2a

This formula allows us to find the values of 'x' directly from the coefficients 'a', 'b', and 'c'. The term inside the square root, (b² – 4ac), is called the discriminant (often denoted by Δ or D). The value of the discriminant tells us about the nature of the roots:

  • If Discriminant > 0: There are two distinct real roots. This means the parabola (the graph of a quadratic equation) intersects the x-axis at two different points.
  • If Discriminant = 0: There is exactly one real root (also called a repeated or double root). The parabola touches the x-axis at exactly one point.
  • If Discriminant < 0: There are two complex conjugate roots. This means the parabola does not intersect the x-axis. Complex roots involve the imaginary unit 'i', where i = √(-1).

How to Use the Calculator

To use the Polynomial Equation Solver:

  1. Enter Coefficient 'a': Input the number that multiplies x². If there's no number explicitly written, it's usually 1. If 'a' is 0, the calculator will treat it as a linear equation.
  2. Enter Coefficient 'b': Input the number that multiplies x.
  3. Enter Coefficient 'c': Input the constant term (the number without any 'x').
  4. Click 'Calculate Roots': The calculator will instantly display the roots of your quadratic equation, indicating whether they are real or complex.

Examples of Quadratic Equations:

Let's look at some examples:

Example 1: Two Distinct Real Roots

Equation: x² – 3x + 2 = 0

  • a = 1
  • b = -3
  • c = 2

Discriminant = (-3)² – 4(1)(2) = 9 – 8 = 1 (which is > 0)

Roots: x₁ = 2, x₂ = 1

Example 2: One Real (Repeated) Root

Equation: x² – 4x + 4 = 0

  • a = 1
  • b = -4
  • c = 4

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

Root: x = 2

Example 3: Two Complex Conjugate Roots

Equation: x² + 2x + 5 = 0

  • a = 1
  • b = 2
  • c = 5

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

Roots: x₁ = -1 + 2i, x₂ = -1 – 2i

This calculator provides a quick and accurate way to solve quadratic equations, helping you understand the nature of their roots based on the coefficients you provide.

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