Quadratic Equation Calculator Step by Step

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Quadratic Equation Calculator (Step-by-Step)

Results

Enter coefficients ‘a’, ‘b’, and ‘c’ to find the solutions.

Understanding the Quadratic Equation

A quadratic equation is a second-degree polynomial equation, meaning it contains at least one term that is squared. The general form of a quadratic equation is:

ax² + bx + c = 0

where:

  • a, b, and c are coefficients (numbers), and
  • a cannot be zero (if a=0, it becomes a linear equation).

The goal is to find the value(s) of x that satisfy this equation. These values are called the roots or solutions of the equation. A quadratic equation can have:

  • Two distinct real roots
  • One real root (a repeated root)
  • Two complex roots (involving imaginary numbers)

The Quadratic Formula

The most common and reliable method for solving quadratic equations is the quadratic formula, derived using a technique called completing the square:

x = [-b ± √(b² – 4ac)] / 2a

The Discriminant (Δ)

A crucial part of the quadratic formula is the expression under the square root sign: b² - 4ac. This 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: There are two distinct real roots.
  • If Δ = 0: There is exactly one real root (a repeated root).
  • If Δ < 0: There are two complex conjugate roots.

Step-by-Step Calculation Process

  1. Identify Coefficients: Determine the values of a, b, and c from your equation (ensure it’s in the standard form ax² + bx + c = 0).
  2. Calculate the Discriminant: Compute Δ = b² - 4ac.
  3. Analyze the Discriminant:
    • If Δ ≥ 0, proceed to calculate real roots.
    • If Δ < 0, calculate complex roots.
  4. Apply the Quadratic Formula: Substitute the values of a, b, c, and the discriminant into the formula to find the value(s) of x.

Use Cases for Quadratic Equations

Quadratic equations appear in various fields:

  • Physics: Modeling projectile motion (e.g., the path of a ball thrown into the air).
  • Engineering: Designing structures, calculating areas, and optimizing performance.
  • Economics: Analyzing cost functions, revenue, and profit maximization.
  • Geometry: Calculating areas and solving problems involving shapes like parabolas.
  • Finance: Modeling growth patterns or investment returns.

function calculateQuadratic() {
var a = parseFloat(document.getElementById(‘coefficientA’).value);
var b = parseFloat(document.getElementById(‘coefficientB’).value);
var c = parseFloat(document.getElementById(‘coefficientC’).value);

var resultsDiv = document.getElementById(‘solution’);
var stepDiv = document.getElementById(‘stepByStepDetails’);
resultsDiv.style.color = ‘#004a99’; // Reset color
stepDiv.innerHTML = ”; // Clear previous steps

// Input validation
if (isNaN(a) || isNaN(b) || isNaN(c)) {
resultsDiv.innerText = ‘Error: Please enter valid numbers for all coefficients.’;
return;
}

if (a === 0) {
resultsDiv.innerText = ‘Error: Coefficient “a” cannot be zero for a quadratic equation.’;
stepDiv.innerHTML = ‘If a = 0, the equation becomes linear: bx + c = 0. The solution is x = -c / b (if b is not zero).’;
return;
}

// Step 1: Display Input Coefficients
stepDiv.innerHTML += ‘Step 1: Identified Coefficients‘;
stepDiv.innerHTML += ‘Equation: ' + a + 'x² + ' + b + 'x + ' + c + ' = 0‘;
stepDiv.innerHTML += ‘a = ‘ + a + ‘, b = ‘ + b + ‘, c = ‘ + c + ”;

// Step 2: Calculate the Discriminant
var discriminant = b * b – 4 * a * c;
stepDiv.innerHTML += ‘Step 2: Calculate the Discriminant (Δ)‘;
stepDiv.innerHTML += ‘Formula: Δ = b² – 4ac’;
stepDiv.innerHTML += ‘Calculation: Δ = (‘ + b + ‘)² – 4 * (‘ + a + ‘) * (‘ + c + ‘) = ‘ + discriminant + ”;

var x1, x2;

// Step 3: Analyze Discriminant and Calculate Roots
stepDiv.innerHTML += ‘Step 3: Analyze Discriminant and Find Roots‘;

if (discriminant > 0) {
// Two distinct real roots
x1 = (-b + Math.sqrt(discriminant)) / (2 * a);
x2 = (-b – Math.sqrt(discriminant)) / (2 * a);

stepDiv.innerHTML += ‘Since Δ (‘ + discriminant + ‘) > 0, there are two distinct real roots.’;
stepDiv.innerHTML += ‘Step 4: Apply the Quadratic Formula‘;
stepDiv.innerHTML += ‘x₁ = [-b + √(Δ)] / 2a = [-(‘ + b + ‘) + √(‘ + discriminant + ‘)] / (2 * ‘ + a + ‘) = ‘ + x1.toFixed(4) + ”;
stepDiv.innerHTML += ‘x₂ = [-b – √(Δ)] / 2a = [-(‘ + b + ‘) – √(‘ + discriminant + ‘)] / (2 * ‘ + a + ‘) = ‘ + x2.toFixed(4) + ”;

resultsDiv.innerText = ‘Solutions: x₁ = ‘ + x1.toFixed(4) + ‘, x₂ = ‘ + x2.toFixed(4);
resultsDiv.style.color = ‘#28a745’; // Success Green

} else if (discriminant === 0) {
// One real root (repeated)
x1 = -b / (2 * a);

stepDiv.innerHTML += ‘Since Δ (‘ + discriminant + ‘) = 0, there is exactly one real root (a repeated root).’;
stepDiv.innerHTML += ‘Step 4: Apply the Quadratic Formula‘;
stepDiv.innerHTML += ‘x = -b / 2a = -(‘ + b + ‘) / (2 * ‘ + a + ‘) = ‘ + x1.toFixed(4) + ”;

resultsDiv.innerText = ‘Solution: x = ‘ + x1.toFixed(4) + ‘ (repeated root)’;
resultsDiv.style.color = ‘#28a745’; // Success Green

} else {
// Two complex roots
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(-discriminant) / (2 * a);

stepDiv.innerHTML += ‘Since Δ (‘ + discriminant + ‘) < 0, there are two complex conjugate roots.';
stepDiv.innerHTML += 'Step 4: Apply the Quadratic Formula for Complex Roots‘;
stepDiv.innerHTML += ‘Real Part = -b / 2a = -(‘ + b + ‘) / (2 * ‘ + a + ‘) = ‘ + realPart.toFixed(4) + ”;
stepDiv.innerHTML += ‘Imaginary Part = √(-Δ) / 2a = √(‘ + (-discriminant) + ‘) / (2 * ‘ + a + ‘) = ‘ + imaginaryPart.toFixed(4) + ”;
stepDiv.innerHTML += ‘x₁ = ‘ + realPart.toFixed(4) + ‘ + i‘ + imaginaryPart.toFixed(4) + ”;
stepDiv.innerHTML += ‘x₂ = ‘ + realPart.toFixed(4) + ‘ – i‘ + imaginaryPart.toFixed(4) + ”;

resultsDiv.innerText = ‘Complex Solutions: x₁ = ‘ + realPart.toFixed(4) + ‘ + i’ + imaginaryPart.toFixed(4) + ‘, x₂ = ‘ + realPart.toFixed(4) + ‘ – i’ + imaginaryPart.toFixed(4);
resultsDiv.style.color = ‘#dc3545’; // Error/Warning Color
}
}

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