Texas Instruments Online Calculators

Quadratic Equation Solver

Results:

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("quadraticResult"); 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., 5 = 0)."; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; resultDiv.innerHTML = "This is a linear equation (a=0). The root 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 = "Two distinct real roots:x₁ = " + x1.toFixed(4) + "x₂ = " + x2.toFixed(4); } else if (discriminant === 0) { var x = -b / (2 * a); resultDiv.innerHTML = "One real root (repeated root):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 = "Two complex conjugate roots:x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "ix₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; border: 1px solid #ddd; border-radius: 8px; padding: 20px; max-width: 500px; margin: 20px auto; box-shadow: 0 4px 8px rgba(0, 0, 0, 0.05); } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; font-size: 1.8em; } .calculator-inputs label { display: block; margin-bottom: 8px; color: #555; font-weight: bold; } .calculator-inputs input[type="number"] { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ccc; border-radius: 5px; font-size: 1em; } .calculator-inputs button { width: 100%; padding: 12px; background-color: #007bff; color: white; border: none; border-radius: 5px; font-size: 1.1em; cursor: pointer; transition: background-color 0.3s ease; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-results { margin-top: 25px; padding-top: 20px; border-top: 1px solid #eee; } .calculator-results h3 { color: #333; margin-bottom: 10px; font-size: 1.4em; } .calculator-results div { background-color: #e9ecef; border: 1px solid #dee2e6; border-radius: 5px; padding: 15px; font-size: 1.1em; color: #333; word-wrap: break-word; }

Understanding Quadratic Equations and How to Solve Them

Quadratic equations are fundamental in mathematics, physics, engineering, and many other fields. They are polynomial equations of the second degree, meaning they contain 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:

  • x represents the unknown variable.
  • a, b, and c are coefficients, with a not equal to zero. If a were zero, the equation would become a linear equation (bx + c = 0).

The Quadratic Formula

The most common and reliable method for solving quadratic equations is by using the quadratic formula. This formula provides the values of x (also known as the roots or solutions) that satisfy the equation. The quadratic formula is:

x = [-b ± sqrt(b² - 4ac)] / 2a

This formula can yield two distinct real roots, one repeated real root, or two complex conjugate roots, depending on the value of the discriminant.

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 discriminant tells us about the nature of the roots:

  • If Δ > 0 (positive): 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 (zero): There is exactly one real root (also called a repeated or double root). The parabola touches the x-axis at exactly one point.
  • If Δ < 0 (negative): There are two complex conjugate roots. The parabola does not intersect the x-axis; its vertex is either entirely above or entirely below the x-axis. Complex roots involve the imaginary unit i, where i = sqrt(-1).

How to Use the Quadratic Equation Solver

Our online Quadratic Equation Solver, similar to functions found on Texas Instruments scientific calculators, simplifies the process of finding the roots. To use it:

  1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember to include their signs (positive or negative).
  2. Enter Values: Input these numerical values into the respective fields: "Coefficient a", "Coefficient b", and "Coefficient c".
  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:

Let's look at a few examples to illustrate the different types of roots:

Example 1: Two Distinct Real Roots

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

  • a = 1
  • b = -3
  • c = 2

Using the calculator with these values will show: x₁ = 2.0000 and x₂ = 1.0000.

Example 2: One Real Root (Repeated)

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

  • a = 1
  • b = -4
  • c = 4

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

Example 3: Two Complex Conjugate Roots

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

  • a = 1
  • b = 2
  • c = 5

Using the calculator with these values will show: x₁ = -1.0000 + 2.0000i and x₂ = -1.0000 - 2.0000i.

This Quadratic Equation Solver is a powerful tool for students, educators, and professionals, providing quick and accurate solutions to one of algebra's most common problems.

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