body {
font-family: 'Segoe UI', Tahoma, Geneva, Verdana, sans-serif;
background-color: #f8f9fa;
color: #333;
line-height: 1.6;
margin: 0;
padding: 20px;
}
.quadratic-calculator-container {
max-width: 800px;
margin: 30px auto;
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
border: 1px solid #e0e0e0;
}
h1, h2 {
color: #004a99;
text-align: center;
margin-bottom: 20px;
}
.input-group {
margin-bottom: 20px;
display: flex;
align-items: center;
gap: 15px;
flex-wrap: wrap;
}
.input-group label {
flex-basis: 150px;
text-align: right;
font-weight: 600;
color: #555;
}
.input-group input[type="number"] {
flex-grow: 1;
padding: 10px 12px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 1rem;
min-width: 120px;
}
.input-group input[type="number"]:focus {
border-color: #004a99;
outline: none;
box-shadow: 0 0 5px rgba(0, 74, 153, 0.3);
}
button {
display: block;
width: 100%;
padding: 12px 20px;
background-color: #004a99;
color: white;
border: none;
border-radius: 4px;
font-size: 1.1rem;
cursor: pointer;
transition: background-color 0.3s ease;
margin-top: 20px;
}
button:hover {
background-color: #003366;
}
.results-container {
margin-top: 30px;
padding: 20px;
background-color: #e7f3ff;
border-radius: 6px;
border: 1px solid #cce0ff;
}
.results-container h2 {
margin-top: 0;
color: #004a99;
}
#solution {
margin-top: 15px;
font-size: 1.1rem;
color: #004a99;
font-weight: bold;
text-align: center;
padding: 10px;
background-color: #ffffff;
border: 1px dashed #004a99;
border-radius: 4px;
}
.step-by-step {
margin-top: 20px;
font-size: 0.95rem;
color: #444;
text-align: left;
}
.step-by-step p {
margin-bottom: 10px;
}
.formula {
font-weight: bold;
color: #003366;
}
.article-section {
margin-top: 40px;
border-top: 1px solid #eee;
padding-top: 30px;
}
.article-section h2 {
text-align: left;
margin-bottom: 15px;
}
.article-section p, .article-section li {
margin-bottom: 15px;
color: #555;
}
.article-section ul {
padding-left: 20px;
}
.article-section code {
background-color: #eef;
padding: 2px 5px;
border-radius: 3px;
font-family: Consolas, Monaco, 'Andale Mono', 'Ubuntu Mono', monospace;
}
@media (max-width: 600px) {
.input-group {
flex-direction: column;
align-items: flex-start;
}
.input-group label {
text-align: left;
flex-basis: auto;
margin-bottom: 5px;
}
.input-group input[type="number"] {
width: calc(100% – 24px);
}
.quadratic-calculator-container {
padding: 20px;
}
}
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:
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, andcare coefficients (numbers), andacannot be zero (ifa=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
- Identify Coefficients: Determine the values of
a,b, andcfrom your equation (ensure it's in the standard form ax² + bx + c = 0). - Calculate the Discriminant: Compute Δ =
b² - 4ac. - Analyze the Discriminant:
- If Δ ≥ 0, proceed to calculate real roots.
- If Δ < 0, calculate complex roots.
- Apply the Quadratic Formula: Substitute the values of
a,b,c, and the discriminant into the formula to find the value(s) ofx.
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.
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
}
}