Best Algebra Calculator

Quadratic Equation Solver

Solve for 'x' in equations of the form ax² + bx + c = 0

Enter coefficients and click 'Calculate Roots'.
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("result"); 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 = "Infinite solutions (0 = 0)"; } else { resultDiv.innerHTML = "No solution (e.g., " + c + " = 0)"; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; resultDiv.innerHTML = "This is a linear equation. One real root: 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); if (discriminant === 0) { resultDiv.innerHTML = "One real root: x = " + x1.toFixed(4); } else { resultDiv.innerHTML = "Two real roots: x1 = " + x1.toFixed(4) + ", x2 = " + x2.toFixed(4); } } else { // Complex roots var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultDiv.innerHTML = "Two complex roots: x1 = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i, x2 = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } }

Understanding the Quadratic Equation Solver

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. One of the most common and important types of equations you'll encounter in algebra is the quadratic equation. Our Quadratic Equation Solver is designed to help you quickly find the roots (solutions) for any quadratic equation in the standard form: ax² + bx + c = 0.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable (usually 'x') is 2. The coefficients 'a', 'b', and 'c' are real numbers, and 'a' cannot be zero. If 'a' were zero, the x² term would vanish, and it would become a linear equation (bx + c = 0).

The Quadratic Formula

The solutions to a quadratic equation are given by the quadratic formula:

x = [-b ± √(b² – 4ac)] / 2a

The term inside the square root, (b² – 4ac), is called the discriminant (often denoted by Δ). The value of the discriminant tells us about the nature of the roots:

  • If Δ > 0: There are two distinct real roots.
  • If Δ = 0: There is exactly one real root (a repeated root).
  • If Δ < 0: There are two complex conjugate roots.

How to Use This Calculator

Using our Quadratic Equation Solver is straightforward:

  1. Identify Coefficients: Look at your quadratic equation and identify the values for 'a', 'b', and 'c'. Remember, if a term is missing, its coefficient is 0 (e.g., for x² – 4 = 0, a=1, b=0, c=-4).
  2. Enter Values: Input the numerical values for 'Coefficient a', 'Coefficient b', and 'Coefficient c' into the respective fields.
  3. Calculate: Click the "Calculate Roots" button.
  4. View Results: The calculator will instantly display the roots of your equation, indicating whether they are real or complex.

Examples of Quadratic Equations

Let's look at a few examples:

Example 1: Two Distinct Real Roots

Equation: x² – 3x + 2 = 0

  • a = 1
  • b = -3
  • c = 2

Using the calculator with these values will yield: x1 = 2.0000, x2 = 1.0000.

Example 2: One Real Root (Repeated)

Equation: x² – 4x + 4 = 0

  • a = 1
  • b = -4
  • c = 4

Using the calculator with these values will yield: x = 2.0000.

Example 3: Two Complex Roots

Equation: x² + 2x + 5 = 0

  • a = 1
  • b = 2
  • c = 5

Using the calculator with these values will yield: x1 = -1.0000 + 2.0000i, x2 = -1.0000 – 2.0000i.

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

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