Math Solver Calculator

Quadratic Equation Solver

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"); if (isNaN(a) || isNaN(b) || isNaN(c)) { resultDiv.innerHTML = "Please enter valid numbers for all coefficients."; return; } if (a === 0) { if (b === 0) { if (c === 0) { resultDiv.innerHTML = "This is a trivial equation (0=0). All real numbers are solutions."; } else { resultDiv.innerHTML = "This is a constant equation (" + c + "=0), which has no solution unless c is 0."; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; resultDiv.innerHTML = "This is a linear equation (a=0).Solution (x): " + x.toFixed(4); } 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); resultDiv.innerHTML = "Two distinct real solutions:x₁ = " + x1.toFixed(4) + "x₂ = " + x2.toFixed(4); } else if (discriminant === 0) { var x = -b / (2 * a); resultDiv.innerHTML = "One real solution (repeated root):x = " + x.toFixed(4); } else { // Discriminant < 0, complex solutions var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultDiv.innerHTML = "No real solutions (complex solutions):x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "ix₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } }

Understanding the Quadratic Equation Solver

A quadratic equation is a polynomial equation of the second degree, meaning it contains at least one term in which the unknown variable is raised to the power of two. The standard form of a quadratic equation is:

ax² + bx + c = 0

Where:

• a, b, and c are coefficients (constants), with a not equal to zero.
• x represents the unknown variable.

Why are Quadratic Equations Important?

Quadratic equations appear in many areas of science, engineering, and mathematics. For example:

• Physics: Describing projectile motion, calculating the trajectory of objects under gravity.
• Engineering: Designing parabolic antennas, analyzing electrical circuits.
• Economics: Modeling supply and demand curves, optimizing profits.
• Mathematics: Solving geometric problems, understanding parabolas.

The Quadratic Formula

The most common method to find the solutions (also known as roots) for a quadratic equation is by using the quadratic formula:

x = [-b ± √(b² – 4ac)] / 2a

This formula provides the values of x that satisfy the equation.

Understanding the Discriminant (b² – 4ac)

The term inside the square root, (b² - 4ac), is called the discriminant. Its value determines the nature of the solutions:

• If Discriminant > 0: There are two distinct real solutions. This means the parabola intersects the x-axis at two different points.
• If Discriminant = 0: There is exactly one real solution (a repeated root). The parabola touches the x-axis at exactly one point.
• If Discriminant < 0: There are no real solutions. Instead, there are two complex (or imaginary) solutions. The parabola does not intersect the x-axis.

How to Use This Calculator

Our Quadratic Equation Solver simplifies the process of finding the roots of any quadratic equation. Simply follow these steps:

1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember, the equation must be in the form ax² + bx + c = 0.
2. Enter Values: Input the numerical values for 'Coefficient a', 'Coefficient b', and 'Coefficient 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 solutions for x, along with an explanation of the type of solutions (real, repeated, or complex).

Examples

Let's look at a few examples:

Example 1: Two Distinct Real Solutions

Equation: x² - 5x + 6 = 0

• a = 1
• b = -5
• c = 6

Using the calculator with these values will yield: x₁ = 3.0000, x₂ = 2.0000.

Example 2: One Real Solution (Repeated Root)

Equation: x² + 4x + 4 = 0

• a = 1
• b = 4
• c = 4

Using the calculator with these values will yield: x = -2.0000.

Example 3: No Real Solutions (Complex Roots)

Equation: x² + x + 1 = 0

• a = 1
• b = 1
• c = 1

Using the calculator with these values will yield: x₁ = -0.5000 + 0.8660i, x₂ = -0.5000 - 0.8660i.

This calculator is a powerful tool for students, educators, and professionals to quickly and accurately solve quadratic equations without manual calculation errors.