Calculator for Math Online

Quadratic Equation Solver

Enter the coefficients for the equation: ax² + bx + c = 0

Results:

Enter coefficients and click "Solve Equation" to see the roots.

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"); var output = ""; if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = "

Error:

Please enter valid numbers for all coefficients."; return; } if (a === 0) { // This is a linear equation: bx + c = 0 if (b === 0) { if (c === 0) { output = "

Result:

This is an identity (0 = 0). Any real number is a solution."; } else { output = "

Result:

This is a contradiction (" + c + " = 0). No solution exists."; } } else { var x = -c / b; output = "

Result (Linear Equation):

Since 'a' is 0, this is a linear equation: " + b + "x + " + c + " = 0The single root is: x = " + x.toFixed(4) + ""; } resultDiv.innerHTML = output; return; } var discriminant = b * b – 4 * a * c; if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); output = "

Result:

The equation has two distinct real roots:"; output += "x₁ = " + x1.toFixed(4) + ""; output += "x₂ = " + x2.toFixed(4) + ""; } else if (discriminant === 0) { var x = -b / (2 * a); output = "

Result:

The equation has one real root (repeated):"; output += "x = " + x.toFixed(4) + ""; } else { var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); output = "

Result:

The equation has two complex conjugate roots:"; output += "x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "i"; output += "x₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } resultDiv.innerHTML = output; }

Understanding Quadratic Equations and How to Solve Them Online

A quadratic equation is a fundamental concept in algebra, representing a polynomial equation of the second degree. It takes the general form:

ax² + bx + c = 0

where 'x' represents an unknown variable, and 'a', 'b', and 'c' are numerical coefficients. The coefficient 'a' cannot be zero; if 'a' were zero, the equation would become a linear equation (bx + c = 0).

Why are Quadratic Equations Important?

Quadratic equations appear in various fields of science, engineering, and economics. For instance:

• Physics: Describing projectile motion, calculating the trajectory of objects under gravity.
• Engineering: Designing structures, analyzing electrical circuits, and optimizing processes.
• Economics: Modeling supply and demand curves, calculating profit maximization.
• Mathematics: Foundation for more advanced algebraic concepts and calculus.

The Quadratic Formula

The most common method to solve a quadratic equation is by using the quadratic formula:

x = [-b ± √(b² – 4ac)] / (2a)

This formula provides the values of 'x' (also known as the roots or solutions) that satisfy the equation.

Understanding the Discriminant (b² – 4ac)

The term inside the square root, Δ = b² – 4ac, is called the discriminant. Its value determines the nature of the roots:

• If Δ > 0: The equation has two distinct real roots. This means there are two different numerical solutions for 'x'.
• If Δ = 0: The equation has exactly one real root (sometimes called a repeated or double root). Both solutions for 'x' are the same.
• If Δ < 0: The equation has two complex conjugate roots. These roots involve imaginary numbers (represented by 'i', where i² = -1) and come in pairs.

How to Use Our Online Quadratic Equation Solver

Our calculator simplifies the process of finding the roots of any quadratic equation. Follow these simple steps:

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., in x² + 5 = 0, b=0).
2. Enter Values: Input the numerical values for 'a', 'b', and 'c' into the respective fields in the calculator above.
3. Solve: Click the "Solve Equation" button.
4. View Results: The calculator will instantly display the roots of your equation, indicating whether they are real or complex.

Examples:

Let's look at a few examples to illustrate how the calculator works:

Example 1: Two Distinct Real Roots

Equation: x² – 5x + 6 = 0

• a = 1
• b = -5
• c = 6

Input these values into the calculator. The result will show: x₁ = 3.0000, x₂ = 2.0000.

Example 2: One Real Root (Repeated)

Equation: x² – 4x + 4 = 0

• a = 1
• b = -4
• c = 4

Input these values. The result will show: x = 2.0000.

Example 3: Two Complex Conjugate Roots

Equation: x² + x + 1 = 0

• a = 1
• b = 1
• c = 1

Input these values. The result will show: x₁ = -0.5000 + 0.8660i, x₂ = -0.5000 – 0.8660i.

Example 4: Linear Equation (a = 0)

Equation: 0x² + 2x + 4 = 0 (which simplifies to 2x + 4 = 0)

• a = 0
• b = 2
• c = 4

Input these values. The calculator will correctly identify it as a linear equation and provide the single root: x = -2.0000.

This online quadratic equation solver is a powerful tool for students, educators, and professionals alike, providing quick and accurate solutions for a wide range of mathematical problems.