Quadratic Equation Calculator with Steps

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Quadratic Equation Calculator with Steps

body {
font-family: ‘Segoe UI’, Tahoma, Geneva, Verdana, sans-serif;
background-color: #f8f9fa;
color: #333;
line-height: 1.6;
margin: 0;
padding: 20px;
display: flex;
justify-content: center;
align-items: flex-start;
min-height: 100vh;
}
.loan-calc-container {
background-color: #ffffff;
padding: 30px;
border-radius: 8px;
box-shadow: 0 4px 15px rgba(0, 0, 0, 0.1);
width: 100%;
max-width: 700px;
box-sizing: border-box;
}
h1, h2 {
color: #004a99;
text-align: center;
margin-bottom: 20px;
}
.input-group {
margin-bottom: 20px;
display: flex;
align-items: center;
flex-wrap: wrap;
}
.input-group label {
flex: 0 0 120px;
margin-right: 15px;
font-weight: 600;
color: #555;
text-align: right;
}
.input-group input[type=”number”],
.input-group input[type=”text”] {
flex: 1 1 200px;
padding: 10px 15px;
border: 1px solid #ccc;
border-radius: 4px;
font-size: 1rem;
box-sizing: border-box;
}
button {
background-color: #004a99;
color: white;
padding: 12px 25px;
border: none;
border-radius: 4px;
cursor: pointer;
font-size: 1rem;
transition: background-color 0.3s ease;
display: block;
width: 100%;
margin-top: 10px;
}
button:hover {
background-color: #003366;
}
#result {
margin-top: 30px;
padding: 20px;
background-color: #e7f3ff;
border-left: 5px solid #004a99;
border-radius: 4px;
font-size: 1.1rem;
text-align: center;
}
#result h3 {
color: #004a99;
margin-top: 0;
}
#result p {
margin: 10px 0;
color: #007bff;
font-weight: bold;
}
#steps {
margin-top: 25px;
padding: 20px;
background-color: #f0f0f0;
border: 1px solid #ddd;
border-radius: 4px;
font-size: 0.95rem;
text-align: left;
}
#steps h3 {
color: #004a99;
margin-top: 0;
text-align: center;
margin-bottom: 15px;
}
#steps p {
margin-bottom: 10px;
color: #333;
}
#steps code {
background-color: #e0e0e0;
padding: 2px 5px;
border-radius: 3px;
}
.error {
color: #dc3545;
font-weight: bold;
text-align: center;
margin-top: 15px;
}
.article-content {
margin-top: 40px;
border-top: 1px solid #eee;
padding-top: 25px;
}
.article-content h2 {
color: #004a99;
text-align: left;
margin-bottom: 15px;
}
.article-content p {
margin-bottom: 15px;
}
.article-content ul, .article-content ol {
margin-left: 20px;
margin-bottom: 15px;
}
.article-content li {
margin-bottom: 8px;
}
.article-content code {
background-color: #f0f0f0;
padding: 2px 6px;
border-radius: 4px;
font-family: Consolas, Monaco, ‘Andale Mono’, ‘Ubuntu Mono’, monospace;
}
@media (max-width: 600px) {
.input-group {
flex-direction: column;
align-items: stretch;
}
.input-group label {
flex: none;
margin-right: 0;
margin-bottom: 5px;
text-align: left;
}
.input-group input[type=”number”],
.input-group input[type=”text”] {
flex: none;
width: 100%;
}
body {
padding: 10px;
}
.loan-calc-container {
padding: 20px;
}
}

Quadratic Equation Calculator

Solve for x in equations of the form ax² + bx + c = 0

Understanding the Quadratic Equation

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

ax² + bx + c = 0

where:

  • a, b, and c are coefficients (constants).
  • x is the variable for which we are solving.
  • a cannot be zero. If a = 0, the equation becomes a linear equation (bx + c = 0).

The Discriminant

The nature of the solutions (roots) of a quadratic equation depends on a value called the discriminant, which is calculated as:

Δ = b² - 4ac

  • If Δ > 0: There are two distinct real solutions.
  • If Δ = 0: There is exactly one real solution (a repeated root).
  • If Δ < 0: There are two complex conjugate solutions.

The Quadratic Formula

The solutions for x are found using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

This formula directly uses the coefficients a, b, and c to find the values of x that satisfy the equation.

How This Calculator Works

This calculator takes the coefficients a, b, and c that you input and:

  1. Checks if a is zero. If it is, it informs you it’s not a quadratic equation and solves it as a linear equation if possible.
  2. Calculates the discriminant (Δ = b² - 4ac) to determine the type of roots.
  3. Applies the quadratic formula to find the real or complex roots.
  4. Displays the calculated roots and shows the step-by-step process, including the discriminant calculation.

Use Cases

Quadratic equations and their solutions have numerous applications in various fields:

  • Physics: Calculating projectile motion (e.g., the path of a ball thrown in the air), determining the time it takes for an object to fall under gravity.
  • Engineering: Designing structures, analyzing electrical circuits.
  • Economics: Modeling cost and revenue functions, finding optimal production levels.
  • Geometry: Finding areas, calculating dimensions of shapes.
  • Mathematics: Understanding parabolas and their properties, solving algebraic problems.

By understanding and solving quadratic equations, we can model and predict phenomena in the real world more effectively.

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(“result”);
var stepsDiv = document.getElementById(“steps”);
var errorDiv = document.getElementById(“error”);

resultDiv.innerHTML = “”;
stepsDiv.innerHTML = “”;
errorDiv.innerHTML = “”;

// Input validation
if (isNaN(a) || isNaN(b) || isNaN(c)) {
errorDiv.innerHTML = “Please enter valid numbers for all coefficients.”;
return;
}

var stepsHTML = “

Calculation Steps

“;
stepsHTML += “Given equation: ax² + bx + c = 0“;
stepsHTML += “Coefficients: a = ” + a + “, b = ” + b + “, c = ” + c + “”;

// Case: a = 0 (Linear Equation)
if (a === 0) {
stepsHTML += “Since a = 0, this is a linear equation: bx + c = 0“;
if (b !== 0) {
var x_linear = -c / b;
stepsHTML += “Solving for x: x = -c / b“;
stepsHTML += “Result: x = ” + x_linear.toFixed(4) + ““;
resultDiv.innerHTML = “

Linear Equation Solution

x = ” + x_linear.toFixed(4) + ““;
} else { // a = 0 and b = 0
stepsHTML += “Since a = 0 and b = 0:”;
if (c === 0) {
stepsHTML += “The equation is 0 = 0, which is true for all values of x (infinite solutions).”;
resultDiv.innerHTML = “

Linear Equation Result

Infinite solutions (0 = 0).”;
} else {
stepsHTML += “The equation is " + c + " = 0, which is false (no solution).”;
resultDiv.innerHTML = “

Linear Equation Result

No solution (" + c + " = 0 is false).”;
}
}
stepsDiv.innerHTML = stepsHTML;
return;
}

// Quadratic Equation Calculations
stepsHTML += “Calculate the discriminant (Δ): Δ = b² - 4ac“;
var discriminant = (b * b) – (4 * a * c);
stepsHTML += “Δ = (” + b + “)² – 4 * (” + a + “) * (” + c + “)”;
stepsHTML += “Δ = ” + (b * b) + ” – ” + (4 * a * c) + “”;
stepsHTML += “Δ = ” + discriminant.toFixed(4) + ““;

var solutions = [];
var resultHTML = “

Quadratic Equation Solutions

“;

if (discriminant > 0) {
stepsHTML += “Since Δ > 0, there are two distinct real solutions.”;
var x1 = (-b + Math.sqrt(discriminant)) / (2 * a);
var x2 = (-b – Math.sqrt(discriminant)) / (2 * a);
solutions.push(x1, x2);

stepsHTML += “Using the quadratic formula: x = [-b ± √Δ] / 2a“;
stepsHTML += “First solution (using +): x₁ = (-" + b + " + √" + discriminant.toFixed(4) + ") / (2 * " + a + ")“;
stepsHTML += “x₁ = (" + (-b) + " + " + Math.sqrt(discriminant).toFixed(4) + ") / " + (2 * a) + "“;
stepsHTML += “x₁ = " + ((-b) + Math.sqrt(discriminant)).toFixed(4) + " / " + (2 * a) + "“;
stepsHTML += “x₁ = ” + x1.toFixed(4) + ““;

stepsHTML += “Second solution (using -): x₂ = (-" + b + " - √" + discriminant.toFixed(4) + ") / (2 * " + a + ")“;
stepsHTML += “x₂ = (" + (-b) + " - " + Math.sqrt(discriminant).toFixed(4) + ") / " + (2 * a) + "“;
stepsHTML += “x₂ = " + ((-b) - Math.sqrt(discriminant)).toFixed(4) + " / " + (2 * a) + "“;
stepsHTML += “x₂ = ” + x2.toFixed(4) + ““;

resultHTML += “x₁ = ” + x1.toFixed(4) + ““;
resultHTML += “x₂ = ” + x2.toFixed(4) + ““;

} else if (discriminant === 0) {
stepsHTML += “Since Δ = 0, there is exactly one real solution (a repeated root).”;
var x = -b / (2 * a);
solutions.push(x);

stepsHTML += “Using the quadratic formula: x = -b / 2a“;
stepsHTML += “x = -(” + b + “) / (2 * ” + a + “)”;
stepsHTML += “x = ” + (-b) + ” / ” + (2 * a) + “”;
stepsHTML += “x = ” + x.toFixed(4) + ““;

resultHTML += “x = ” + x.toFixed(4) + “ (repeated root)”;

} else { // discriminant < 0
stepsHTML += "Since Δ < 0, there are two complex conjugate solutions.";
var realPart = -b / (2 * a);
var imaginaryPart = Math.sqrt(-discriminant) / (2 * a);
solutions.push(realPart + " + " + imaginaryPart + "i", realPart + " – " + imaginaryPart + "i");

stepsHTML += "Using the quadratic formula: x = [-b ± i√(-Δ)] / 2a“;
stepsHTML += “Real part: -b / 2a“;
stepsHTML += “Real part = -” + b + ” / (2 * ” + a + “) = ” + realPart.toFixed(4) + “”;
stepsHTML += “Imaginary part: √(-Δ) / 2a“;
stepsHTML += “Imaginary part = √(” + (-discriminant).toFixed(4) + “) / ” + (2 * a) + ” = ” + Math.sqrt(-discriminant).toFixed(4) + ” / ” + (2 * a) + ” = ” + imaginaryPart.toFixed(4) + “”;

stepsHTML += “First complex solution: x₁ = ” + realPart.toFixed(4) + ” + ” + imaginaryPart.toFixed(4) + “i“;
stepsHTML += “Second complex solution: x₂ = ” + realPart.toFixed(4) + ” – ” + imaginaryPart.toFixed(4) + “i“;

resultHTML += “x₁ = ” + realPart.toFixed(4) + ” + ” + imaginaryPart.toFixed(4) + “i“;
resultHTML += “x₂ = ” + realPart.toFixed(4) + ” – ” + imaginaryPart.toFixed(4) + “i“;
}

resultDiv.innerHTML = resultHTML;
stepsDiv.innerHTML = stepsHTML;
}

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