Solving Polynomials Calculator

Quadratic Polynomial Solver

Enter the coefficients for a quadratic equation of the form ax² + bx + c = 0 to find its roots.

function calculatePolynomial() { 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("polynomialResult"); 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) { // Handle linear equation or degenerate cases 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 is false (no solution)."; } } else { // bx + c = 0 => x = -c/b var x = -c / b; resultDiv.innerHTML = "This is a linear equation (a=0). The solution is x = " + x.toFixed(4) + ""; } 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); resultDiv.innerHTML = "The equation has two distinct real roots:" + "x₁ = " + x1.toFixed(4) + "" + "x₂ = " + x2.toFixed(4) + ""; } else if (discriminant === 0) { var x = -b / (2 * a); resultDiv.innerHTML = "The equation has one real root (repeated):" + "x = " + x.toFixed(4) + ""; } else { // discriminant < 0, complex roots var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultDiv.innerHTML = "The equation has two complex conjugate roots:" + "x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i" + "x₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } }

Understanding and Solving Quadratic Polynomials

A polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. Among the various types of polynomials, quadratic polynomials are particularly common and fundamental in algebra, physics, engineering, and many other fields.

What is a Quadratic Polynomial?

A quadratic polynomial is a polynomial of degree 2, meaning the highest exponent of the variable in the expression is 2. Its standard form is:

ax² + bx + c = 0

Where:

• a, b, and c are coefficients (real numbers).
• a cannot be zero (if a=0, it becomes a linear equation: bx + c = 0).
• x is the variable.

The solutions to a quadratic equation are called its "roots" or "zeros." These are the values of x for which the equation holds true.

The Quadratic Formula

The most common and reliable method for finding the roots of any quadratic equation is the quadratic formula. This formula directly provides the values of x based on the coefficients a, b, and c:

x = [-b ± sqrt(b² - 4ac)] / 2a

The "±" symbol indicates that there are generally two solutions: one where you add the square root term and one where you subtract it.

The Discriminant (b² – 4ac)

The term inside the square root, (b² - 4ac), is called the "discriminant" (often denoted by the Greek letter Delta, Δ). The value of the discriminant determines 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 root or a double root). The parabola touches the x-axis at exactly one point.
• If discriminant < 0: There are two complex conjugate roots. The parabola does not intersect the x-axis at all. Complex roots involve the imaginary unit i, where i = sqrt(-1).

How to Use the Calculator

To use the Quadratic Polynomial Solver, simply input the numerical values for the coefficients a, b, and c into their respective fields. Then, click the "Calculate Roots" button. The calculator will instantly display the roots of your quadratic equation, indicating whether they are real or complex.

Examples:

Example 1: Two Distinct Real Roots

Consider the equation: x² - 5x + 6 = 0

• a = 1
• b = -5
• c = 6

Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1 (which is > 0)

Roots: x = [5 ± sqrt(1)] / 2(1)

x₁ = (5 + 1) / 2 = 3

x₂ = (5 - 1) / 2 = 2

(Enter a=1, b=-5, c=6 into the calculator to verify.)

Example 2: One Real (Repeated) Root

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

• a = 1
• b = -4
• c = 4

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

Roots: x = [4 ± sqrt(0)] / 2(1)

x = 4 / 2 = 2

(Enter a=1, b=-4, c=4 into the calculator to verify.)

Example 3: Two Complex Conjugate Roots

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

• a = 1
• b = 2
• c = 5

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

Roots: x = [-2 ± sqrt(-16)] / 2(1)

x = [-2 ± 4i] / 2

x₁ = -1 + 2i

x₂ = -1 - 2i

(Enter a=1, b=2, c=5 into the calculator to verify.)

This calculator provides a quick and accurate way to find the roots of any quadratic polynomial, making it a valuable tool for students, educators, and professionals alike.