How to Calculate Quadratic Formula

Quadratic Formula Calculator

Enter the coefficients a, b, and c for a quadratic equation in the form ax² + bx + c = 0.

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 = "Infinite solutions (0 = 0)."; } else { resultDiv.innerHTML = "No solution (e.g., 5 = 0)."; } } else { // Linear equation: bx + c = 0 => x = -c/b var x = -c / b; resultDiv.innerHTML = "This is a linear equation (a=0).The root (x) is: " + x.toFixed(4) + ""; } return; } var discriminant = b * b – 4 * a * c; var x1, x2; if (discriminant > 0) { x1 = (-b + Math.sqrt(discriminant)) / (2 * a); x2 = (-b – Math.sqrt(discriminant)) / (2 * a); resultDiv.innerHTML = "Two distinct real roots:x₁ = " + x1.toFixed(4) + "x₂ = " + x2.toFixed(4) + ""; } else if (discriminant === 0) { x1 = -b / (2 * a); resultDiv.innerHTML = "One real root (repeated):x = " + x1.toFixed(4) + ""; } else { // discriminant < 0, complex roots var realPart = -b / (2 * a); var imaginaryPart = Math.sqrt(Math.abs(discriminant)) / (2 * a); resultDiv.innerHTML = "Two complex conjugate roots:x₁ = " + realPart.toFixed(4) + " + " + imaginaryPart.toFixed(4) + "ix₂ = " + realPart.toFixed(4) + " – " + imaginaryPart.toFixed(4) + "i"; } }

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to solve any quadratic equation, which is an equation of the second degree. A standard quadratic equation is expressed in the form:

ax² + bx + c = 0

where 'a', 'b', and 'c' are coefficients, and 'a' cannot be zero. If 'a' were zero, the equation would become a linear equation (bx + c = 0). The solutions to a quadratic equation are also known as its "roots" or "zeros," as they represent the x-values where the parabola (the graph of a quadratic function) intersects the x-axis.

The Formula Itself

The quadratic formula provides the values for 'x' directly:

x = [-b ± √(b² – 4ac)] / 2a

This formula allows you to find up to two possible values for 'x' that satisfy the equation. The "±" symbol indicates that there are two potential solutions: one where you add the square root term, and one where you subtract it.

The Discriminant (b² – 4ac)

A crucial part of the quadratic formula is the expression under the square root, known as the discriminant (Δ):

Δ = b² – 4ac

The value of the discriminant tells us about the nature of the roots:

• If Δ > 0 (Discriminant is positive): There are two distinct real roots. This means the parabola intersects the x-axis at two different points.
• If Δ = 0 (Discriminant is zero): There is exactly one real root (sometimes called a repeated or double root). The parabola touches the x-axis at exactly one point.
• If Δ < 0 (Discriminant is negative): There are two complex conjugate roots. The parabola does not intersect the x-axis at all. These roots involve the imaginary unit 'i' (where i = √-1).

How to Use the Calculator

To use the Quadratic Formula Calculator, simply identify the coefficients 'a', 'b', and 'c' from your quadratic equation (make sure it's in the standard form ax² + bx + c = 0).

1. Enter 'a': Input the coefficient of the x² term.
2. Enter 'b': Input the coefficient of the x term.
3. Enter 'c': Input the constant term.
4. Click "Calculate Roots": 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 – 3x + 2 = 0
Here, a = 1, b = -3, c = 2.
Discriminant = (-3)² – 4(1)(2) = 9 – 8 = 1 (positive)
Roots: x₁ = [-(-3) + √1] / (2*1) = (3 + 1) / 2 = 2
        x₂ = [-(-3) – √1] / (2*1) = (3 – 1) / 2 = 1
(Enter a=1, b=-3, c=2 into the calculator to see these results.)

Example 2: One Real (Repeated) Root

Equation: x² + 4x + 4 = 0
Here, a = 1, b = 4, c = 4.
Discriminant = (4)² – 4(1)(4) = 16 – 16 = 0 (zero)
Root: x = [-4 + √0] / (2*1) = -4 / 2 = -2
(Enter a=1, b=4, c=4 into the calculator to see this result.)

Example 3: Two Complex Conjugate Roots

Equation: x² + x + 1 = 0
Here, a = 1, b = 1, c = 1.
Discriminant = (1)² – 4(1)(1) = 1 – 4 = -3 (negative)
Roots: x₁ = [-1 + √-3] / (2*1) = -0.5 + 0.8660i
        x₂ = [-1 – √-3] / (2*1) = -0.5 – 0.8660i
(Enter a=1, b=1, c=1 into the calculator to see these results.)

The quadratic formula is a fundamental concept in algebra and has wide applications in various fields, including physics, engineering, and economics, wherever parabolic trajectories or relationships are modeled.