Quadratic Equation Calculator

Quadratic Equation Solver

Solve for x in the equation: ax² + bx + c = 0

function calculateQuadratic() { 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("quadraticResult"); resultDiv.style.color = '#333'; // Reset color for new results if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients."; resultDiv.style.color = 'red'; return; } if (a === 0) { if (b === 0) { if (c === 0) { resultDiv.innerHTML = "This is the trivial equation 0 = 0, which has infinite solutions."; } else { resultDiv.innerHTML = "This is the equation " + c + " = 0, which has no solution."; } } else { // Linear equation: bx + c = 0 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 (or two identical real roots):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) + "ix₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } }

Understanding Quadratic Equations and Their Solutions

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

  • a, b, and c are coefficients (constants), with a ≠ 0.
  • x represents the unknown variable.

Quadratic equations are fundamental in mathematics and appear in various fields, including physics (e.g., projectile motion), engineering (e.g., designing parabolic antennas), economics, and computer graphics. Solving a quadratic equation means finding the values of x that satisfy the equation, also known as the roots or zeros of the equation.

The Quadratic Formula

The most common and reliable method for solving any quadratic equation is using the quadratic formula:

x = [-b ± √(b² – 4ac)] / 2a

This formula directly provides the values of x once the coefficients a, b, and c are known.

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 Δ > 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 Δ = 0: There is exactly one real root (sometimes called a repeated or double root). The parabola touches the x-axis at exactly one point.
  • If Δ < 0: There are two complex conjugate roots. The parabola does not intersect the x-axis. These roots involve the imaginary unit 'i' (where i² = -1).

How to Use the Quadratic Equation Solver

Our Quadratic Equation Solver simplifies the process of finding the roots for any quadratic equation. Follow these steps:

  1. Identify Coefficients: Ensure your equation is in the standard form ax² + bx + c = 0.
  2. Enter Values: Input the numerical values for coefficients 'a', 'b', and 'c' into the respective fields in the calculator above.
  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 to illustrate the different types of roots:

Example 1: Two Distinct Real Roots

Equation: x² - 5x + 6 = 0

  • a = 1
  • b = -5
  • c = 6

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

Roots: x₁ = 3, x₂ = 2

(Try entering a=1, b=-5, c=6 into the calculator.)

Example 2: One Real Root

Equation: x² + 4x + 4 = 0

  • a = 1
  • b = 4
  • c = 4

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

Root: x = -2

(Try entering a=1, b=4, c=4 into the calculator.)

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

(Try entering a=1, b=2, c=5 into the calculator.)

This calculator is a handy tool for students, engineers, and anyone needing to quickly and accurately solve quadratic equations without manual calculation.

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