Math Problem Calculator

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Quadratic Equation Solver

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) { resultDiv.innerHTML = "Coefficient 'a' cannot be zero for a quadratic equation. For a linear equation, x = -c/b."; return; } var discriminant = b * b – 4 * a * c; var x1, x2; if (discriminant > 0) { x1 = (-b + Math.sqrt(discriminant)) / (2 * a); 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) { x1 = -b / (2 * a); resultDiv.innerHTML = "The equation has one real root (or two identical real roots):x = " + x1.toFixed(4) + ""; } else { // discriminant < 0 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"; } } .calculator-container { font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif; background-color: #f9f9f9; border: 1px solid #ddd; border-radius: 8px; padding: 25px; max-width: 500px; margin: 30px auto; box-shadow: 0 4px 12px rgba(0, 0, 0, 0.08); } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 25px; font-size: 1.8em; } .calculator-content { display: flex; flex-direction: column; gap: 15px; } .input-group { display: flex; flex-direction: column; } .input-group label { margin-bottom: 8px; color: #555; font-size: 1em; font-weight: bold; } .input-group input[type="number"] { padding: 12px; border: 1px solid #ccc; border-radius: 5px; font-size: 1.1em; width: 100%; box-sizing: border-box; transition: border-color 0.3s ease; } .input-group input[type="number"]:focus { border-color: #007bff; outline: none; box-shadow: 0 0 5px rgba(0, 123, 255, 0.3); } .calculate-button { background-color: #007bff; color: white; padding: 14px 20px; border: none; border-radius: 5px; cursor: pointer; font-size: 1.1em; font-weight: bold; margin-top: 15px; transition: background-color 0.3s ease, transform 0.2s ease; } .calculate-button:hover { background-color: #0056b3; transform: translateY(-2px); } .calculate-button:active { transform: translateY(0); } .result-area { background-color: #e9f7ef; border: 1px solid #d4edda; border-radius: 5px; padding: 15px; margin-top: 20px; color: #155724; font-size: 1.1em; line-height: 1.6; word-wrap: break-word; } .result-area p { margin: 5px 0; } .result-area .error { color: #721c24; background-color: #f8d7da; border-color: #f5c6cb; padding: 10px; border-radius: 5px; }

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 written as:

ax² + bx + c = 0

Where:

  • a, b, and c are coefficients, with a not equal to zero.
  • x represents the unknown variable.

Solving a quadratic equation means finding the values of x that satisfy the equation. These values are also known as the roots or zeros of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, and using the quadratic formula.

The Quadratic Formula

The most universal method for solving any quadratic equation is the quadratic formula. It provides a direct way to find the roots, regardless of whether they are real or complex. The formula is:

x = [-b ± sqrt(b² - 4ac)] / 2a

The term (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 (positive discriminant): There are two distinct real roots. This means the parabola (the graph of the quadratic equation) intersects the x-axis at two different points.
  • If Δ = 0 (zero discriminant): 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 (negative discriminant): There are two complex conjugate roots. The parabola does not intersect the x-axis.

How to Use the Quadratic Equation Solver

Our Quadratic Equation Solver simplifies the process of finding these roots. To use it:

  1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember that if a term is missing, its coefficient is 0 (e.g., if there's no 'bx' term, b=0).
  2. Enter Values: Input the numerical values for 'Coefficient a', 'Coefficient b', and 'Coefficient 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 and Their Solutions

Let's look at a few examples to illustrate how the calculator works:

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

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

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

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

Example 3: Two Complex Conjugate Roots

Equation: x² + x + 1 = 0

  • a = 1
  • b = 1
  • c = 1

Using the calculator with these values will yield:

x₁ = -0.5000 + 0.8660i

x₂ = -0.5000 - 0.8660i

(Discriminant: (1)² – 4*1*1 = 1 – 4 = -3, which is < 0)

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

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