Algebra Step by Step Calculator

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Quadratic Equation Solver (ax² + bx + c = 0)







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('resultOutput'); 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) { resultDiv.innerHTML = 'Coefficient \'a\' cannot be zero for a quadratic equation. This would be a linear equation.'; return; } var outputHTML = '

Step-by-Step Solution:

'; outputHTML += 'Given quadratic equation: ' + a + 'x² + ' + b + 'x + ' + c + ' = 0'; outputHTML += 'We use the quadratic formula: x = [-b ± √(b² – 4ac)] / (2a)'; // Step 1: Calculate the Discriminant (Δ) var discriminant = (b * b) – (4 * a * c); outputHTML += 'Step 1: Calculate the Discriminant (Δ)'; outputHTML += 'Δ = b² – 4ac'; outputHTML += 'Δ = (' + b + ')² – 4 * (' + a + ') * (' + c + ')'; outputHTML += 'Δ = ' + (b * b) + ' – ' + (4 * a * c) + "; outputHTML += 'Δ = ' + discriminant.toFixed(4) + "; // Step 2: Determine the nature of the roots and calculate them if (discriminant > 0) { outputHTML += 'Step 2: Since Δ > 0, there are two distinct real roots.'; var sqrtDiscriminant = Math.sqrt(discriminant); var x1 = (-b + sqrtDiscriminant) / (2 * a); var x2 = (-b – sqrtDiscriminant) / (2 * a); outputHTML += 'x₁ = [-b + √Δ] / (2a)'; outputHTML += 'x₁ = [' + (-b) + ' + √(' + discriminant.toFixed(4) + ')] / (2 * ' + a + ')'; outputHTML += 'x₁ = [' + (-b) + ' + ' + sqrtDiscriminant.toFixed(4) + '] / ' + (2 * a) + "; outputHTML += 'x₁ = ' + ((-b + sqrtDiscriminant)).toFixed(4) + ' / ' + (2 * a) + "; outputHTML += 'x₁ = ' + x1.toFixed(4) + ''; outputHTML += 'x₂ = [-b – √Δ] / (2a)'; outputHTML += 'x₂ = [' + (-b) + ' – √(' + discriminant.toFixed(4) + ')] / (2 * ' + a + ')'; outputHTML += 'x₂ = [' + (-b) + ' – ' + sqrtDiscriminant.toFixed(4) + '] / ' + (2 * a) + "; outputHTML += 'x₂ = ' + ((-b – sqrtDiscriminant)).toFixed(4) + ' / ' + (2 * a) + "; outputHTML += 'x₂ = ' + x2.toFixed(4) + ''; outputHTML += 'The two real roots are approximately x₁ = ' + x1.toFixed(4) + ' and x₂ = ' + x2.toFixed(4) + '.'; } else if (discriminant === 0) { outputHTML += 'Step 2: Since Δ = 0, there is exactly one real root (a repeated root).'; var x = -b / (2 * a); outputHTML += 'x = -b / (2a)'; outputHTML += 'x = ' + (-b) + ' / (2 * ' + a + ')'; outputHTML += 'x = ' + (-b) + ' / ' + (2 * a) + "; outputHTML += 'x = ' + x.toFixed(4) + ''; outputHTML += 'The single real root is approximately x = ' + x.toFixed(4) + '.'; } else { // discriminant < 0 outputHTML += 'Step 2: Since Δ < 0, there are two complex conjugate roots.'; var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); outputHTML += 'x = [-b ± i√|Δ|] / (2a)'; outputHTML += 'Real Part = -b / (2a)'; outputHTML += 'Real Part = ' + (-b) + ' / (2 * ' + a + ')'; outputHTML += 'Real Part = ' + realPart.toFixed(4) + "; outputHTML += 'Imaginary Part = √|Δ| / (2a)'; outputHTML += 'Imaginary Part = √|' + discriminant.toFixed(4) + '| / (2 * ' + a + ')'; outputHTML += 'Imaginary Part = ' + Math.sqrt(Math.abs(discriminant)).toFixed(4) + ' / ' + (2 * a) + "; outputHTML += 'Imaginary Part = ' + imaginaryPart.toFixed(4) + "; outputHTML += 'x₁ = ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i'; outputHTML += 'x₂ = ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i'; outputHTML += 'The two complex conjugate roots are approximately x₁ = ' + realPart.toFixed(4) + ' + ' + imaginaryPart.toFixed(4) + 'i and x₂ = ' + realPart.toFixed(4) + ' – ' + imaginaryPart.toFixed(4) + 'i.'; } resultDiv.innerHTML = outputHTML; } .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-inputs label { display: inline-block; width: 120px; margin-bottom: 8px; font-weight: bold; } .calculator-inputs input[type="number"] { width: calc(100% – 130px); padding: 8px; margin-bottom: 10px; border: 1px solid #ccc; border-radius: 4px; } .calculator-inputs button { display: block; width: 100%; padding: 10px 15px; background-color: #007bff; color: white; border: none; border-radius: 4px; font-size: 16px; cursor: pointer; margin-top: 15px; } .calculator-inputs button:hover { background-color: #0056b3; } .calculator-output { margin-top: 20px; padding: 15px; background-color: #e9ecef; border: 1px solid #ced4da; border-radius: 4px; } .calculator-output h3 { color: #007bff; margin-top: 0; } .calculator-output p { margin-bottom: 8px; line-height: 1.5; } .calculator-output strong { color: #333; }

Understanding the Quadratic Equation and Its Solutions

Algebra is a fundamental branch of mathematics that deals with symbols and the rules for manipulating these symbols. One of the most common and important types of equations you'll encounter in algebra is the quadratic equation. This calculator provides a step-by-step solution for any quadratic equation in its standard form.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the variable is squared, but no term with a higher power. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

  • x represents the unknown variable.
  • a, b, and c are coefficients, which are known numbers.
  • a cannot be equal to 0 (if a=0, the equation becomes linear: bx + c = 0).

The solutions to a quadratic equation are also known as its "roots" or "zeros." These are the values of 'x' that make the equation true.

The Quadratic Formula: Your Key to Solving

While some quadratic equations can be solved by factoring or completing the square, the most universal method is using the quadratic formula. This formula directly provides the values of 'x' for any quadratic equation:

x = [-b ± √(b² – 4ac)] / (2a)

Let's break down the components of this formula:

  • -b: The negative of the coefficient 'b'.
  • ±: This symbol indicates that there will generally be two solutions: one where you add the square root term, and one where you subtract it.
  • √(b² – 4ac): This is the square root of the discriminant.
  • 2a: Twice the coefficient 'a'.

Understanding the Discriminant (Δ)

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 tells us about the nature of the roots without actually solving the entire equation:

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

How to Use the Calculator

Our Quadratic Equation Solver simplifies the process:

  1. Identify Coefficients: Look at your quadratic equation and identify the values for 'a', 'b', and 'c'. Remember, if a term is missing, its coefficient is 0 (e.g., if you have x² + 5 = 0, then b=0).
  2. Enter Values: Input these numerical values into the respective fields for Coefficient 'a', 'b', and 'c'.
  3. Click "Solve Equation": The calculator will then display the step-by-step process, starting with the calculation of the discriminant and then proceeding to find the roots based on its value.

Examples of Quadratic Equations

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

Example 1: Two Distinct Real Roots

Equation: x² – 5x + 6 = 0

  • a = 1
  • b = -5
  • c = 6

Using the calculator with these values will show:

  • Δ = (-5)² – 4(1)(6) = 25 – 24 = 1
  • Since Δ > 0, there are two real roots.
  • x₁ = [5 + √1] / 2 = (5 + 1) / 2 = 3
  • x₂ = [5 – √1] / 2 = (5 – 1) / 2 = 2
  • Solutions: x = 3, x = 2

Example 2: One Real Root (Repeated)

Equation: x² – 4x + 4 = 0

  • a = 1
  • b = -4
  • c = 4

Using the calculator with these values will show:

  • Δ = (-4)² – 4(1)(4) = 16 – 16 = 0
  • Since Δ = 0, there is one real root.
  • x = [4 ± √0] / 2 = 4 / 2 = 2
  • Solution: x = 2

Example 3: Two Complex Conjugate Roots

Equation: x² + 2x + 5 = 0

  • a = 1
  • b = 2
  • c = 5

Using the calculator with these values will show:

  • Δ = (2)² – 4(1)(5) = 4 – 20 = -16
  • Since Δ < 0, there are two complex conjugate roots.
  • x = [-2 ± √(-16)] / 2 = [-2 ± 4i] / 2
  • x₁ = -1 + 2i
  • x₂ = -1 – 2i
  • Solutions: x = -1 + 2i, x = -1 – 2i

This calculator is a powerful tool for students and anyone needing to quickly and accurately solve quadratic equations while understanding the underlying steps.

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