Standard Algorithm Calculator

Quadratic Formula Solver

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"); 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. This would be a linear equation."; 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)).toFixed(4); var imaginaryPart = (Math.sqrt(Math.abs(discriminant)) / (2 * a)).toFixed(4); resultDiv.innerHTML = "The equation has two complex roots:" + "x₁ = " + realPart + " + " + imaginaryPart + "i" + "x₂ = " + realPart + " – " + imaginaryPart + "i"; } } .calculator-container { background-color: #f9f9f9; border: 1px solid #ddd; padding: 20px; border-radius: 8px; max-width: 500px; margin: 20px auto; font-family: Arial, sans-serif; } .calculator-container h2 { text-align: center; color: #333; margin-bottom: 20px; } .calculator-inputs label { display: block; margin-bottom: 5px; color: #555; } .calculator-inputs input[type="number"] { width: calc(100% – 22px); padding: 10px; margin-bottom: 15px; border: 1px solid #ccc; border-radius: 4px; box-sizing: border-box; } .calculator-inputs button { background-color: #007bff; color: white; padding: 12px 20px; border: none; border-radius: 4px; cursor: pointer; font-size: 16px; width: 100%; box-sizing: border-box; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-result { margin-top: 20px; padding: 15px; border: 1px solid #e0e0e0; border-radius: 4px; background-color: #eaf4ff; min-height: 50px; } .calculator-result p { margin: 5px 0; color: #333; line-height: 1.5; } .calculator-result p.error { color: #d9534f; font-weight: bold; }

Understanding the Quadratic Formula and Its Solver

The quadratic formula is a fundamental tool in algebra used to solve quadratic equations, which are polynomial equations of the second degree. A standard quadratic equation is expressed in the form:

ax² + bx + c = 0

where 'a', 'b', and 'c' are coefficients, and 'x' represents the unknown variable. The coefficient 'a' cannot be zero, as that would reduce the equation to a linear one.

What is the Quadratic Formula?

The quadratic formula provides the values of 'x' that satisfy the equation. It is given by:

x = [-b ± sqrt(b² - 4ac)] / 2a

This formula allows you to find the roots (or solutions) of any quadratic equation, regardless of whether they are real or complex numbers.

Components of the Formula

  • a, b, c: These are the numerical coefficients of the quadratic equation. 'a' is the coefficient of x², 'b' is the coefficient of x, and 'c' is the constant term.
  • Discriminant (b² – 4ac): The term inside the square root is called the discriminant, often denoted by the Greek letter Delta (Δ). Its value determines the nature of the roots:
    • If Δ > 0: There are two distinct real roots.
    • If Δ = 0: There is exactly one real root (sometimes called a repeated root or two identical real roots).
    • If Δ < 0: There are two complex conjugate roots.

How to Use the Quadratic Formula Solver

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

  1. Identify Coefficients: Look at your quadratic equation and identify the values for 'a', 'b', and 'c'. For example, in the equation 2x² + 5x - 3 = 0, 'a' would be 2, 'b' would be 5, and 'c' would be -3.
  2. Enter Values: Input these numerical values into the respective fields for 'Coefficient 'a", 'Coefficient 'b", and 'Coefficient 'c" 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

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, the discriminant (b² – 4ac) would be (-3)² – 4(1)(2) = 9 – 8 = 1. Since 1 > 0, there are two distinct real roots.

The calculator would output:
x₁ = 2.0000
x₂ = 1.0000

Example 2: One Real Root (Repeated)

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

  • a = 1
  • b = -4
  • c = 4

Here, the discriminant (b² – 4ac) is (-4)² – 4(1)(4) = 16 – 16 = 0. Since the discriminant is 0, there is one real root.

The calculator would output:
x = 2.0000

Example 3: Two Complex Roots

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

  • a = 1
  • b = 2
  • c = 5

In this case, the discriminant (b² – 4ac) is (2)² – 4(1)(5) = 4 – 20 = -16. Since -16 < 0, there are two complex roots.

The calculator would output:
x₁ = -1.0000 + 2.0000i
x₂ = -1.0000 – 2.0000i

This calculator is an invaluable tool for students, educators, and professionals who frequently encounter quadratic equations in mathematics, physics, engineering, and other scientific fields.

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