Pre Calc Calculator

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Quadratic Equation Solver

Enter the coefficients a, b, and c for a quadratic equation in the form ax² + bx + c = 0 to find its 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("quadraticResult"); 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) { if (b === 0) { if (c === 0) { resultDiv.innerHTML = "This is the trivial equation 0 = 0, which has infinitely many solutions."; } else { resultDiv.innerHTML = "This is the equation " + c + " = 0, which has no solution."; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; resultDiv.innerHTML = "This is a linear equation (a=0). The single root is: x = " + x.toFixed(4) + ""; } return; } var discriminant = b * b – 4 * a * c; if (discriminant >= 0) { // Real roots var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); if (discriminant === 0) { resultDiv.innerHTML = "The equation has one real (repeated) root:x = " + x1.toFixed(4) + ""; } else { resultDiv.innerHTML = "The equation has two distinct 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 = "The equation has two complex conjugate roots:x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "ix₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 600px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { color: #333; text-align: center; margin-bottom: 20px; } .calculator-container p { margin-bottom: 15px; line-height: 1.6; } .calculator-input-group { margin-bottom: 15px; } .calculator-input-group label { display: block; margin-bottom: 5px; font-weight: bold; color: #555; } .calculator-input-group input[type="number"] { width: calc(100% – 22px); padding: 10px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-container button { display: block; width: 100%; padding: 12px 20px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 16px; cursor: pointer; transition: background-color 0.3s ease; } .calculator-container button:hover { background-color: #0056b3; } .calculator-result { margin-top: 20px; padding: 15px; background-color: #e9ecef; border: 1px solid #dee2e6; border-radius: 4px; min-height: 50px; } .calculator-result p { margin: 5px 0; font-size: 1.1em; color: #333; } .calculator-result strong { color: #0056b3; } .calculator-result .error { color: #dc3545; font-weight: bold; }

Understanding the Quadratic Equation and Its Roots

In pre-calculus, one of the fundamental concepts you'll encounter is the quadratic equation. 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 expressed as:

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).

What are Roots?

The "roots" (also known as "solutions" or "zeros") of a quadratic equation are the values of x that satisfy the equation, meaning when you substitute these values into the equation, the equation holds true (it equals zero). Graphically, these roots represent the x-intercepts of the parabola that the quadratic equation forms when plotted on a coordinate plane.

The Quadratic Formula

While some quadratic equations can be solved by factoring or completing the square, the quadratic formula is a universal method that works for all quadratic equations. It provides a direct way to find the roots:

x = [-b ± √(b² - 4ac)] / 2a

This formula allows us to calculate two potential roots, often denoted as x₁ and x₂, due to the "±" (plus or minus) sign.

The Discriminant (b² – 4ac)

A crucial part of the quadratic formula is the expression under the square root, b² - 4ac. This is called the "discriminant" (often denoted by the Greek letter Delta, Δ). The value of the discriminant tells us about the nature of the roots:

  • If Δ > 0: There are two distinct real roots. This means the parabola intersects the x-axis at two different points.
  • If Δ = 0: There is exactly one real root (a repeated root). This means the parabola touches the x-axis at exactly one point (its vertex).
  • If Δ < 0: There are two complex conjugate roots. This means the parabola does not intersect the x-axis at all. The roots involve the imaginary unit i, where i = √(-1).

How to Use the Quadratic Equation Solver

Our calculator simplifies the process of finding these roots. Simply input the numerical values for the coefficients a, b, and c from your quadratic equation ax² + bx + c = 0 into the respective fields. Click "Calculate Roots," and the calculator will instantly provide the solutions, indicating whether they are real or complex.

Example Scenarios:

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 yield: x₁ = 2 and x₂ = 1.

Example 2: One Real (Repeated) Root

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

  • a = 1
  • b = -4
  • c = 4

Using the calculator with these values will yield: x = 2.

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 yield: x₁ = -1 + 2i and x₂ = -1 - 2i.

This calculator is a valuable tool for students and professionals alike, helping to quickly verify solutions and deepen understanding of quadratic equations in pre-calculus and beyond.

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