Quadratic Equations Calculator

Quadratic Equation Solver

Find the roots for equations in the form: ax² + bx + c = 0

Calculation Results:

Discriminant (D):

Nature of Roots:

Roots (x):

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Understanding Quadratic Equations

A quadratic equation is a second-order polynomial equation in a single variable x, with the generic form ax² + bx + c = 0. The values of a, b, and c are constants where a cannot be zero. These equations represent parabolas when graphed and are fundamental to algebra, physics, and engineering.

The Quadratic Formula

To find the values of x (the roots), we use the quadratic formula:

x = [-b ± √(b² – 4ac)] / 2a

The Importance of the Discriminant (D)

The term inside the square root, b² – 4ac, is known as the discriminant. It determines the nature of the roots:

• D > 0: There are two distinct real roots. The parabola crosses the x-axis twice.
• D = 0: There is exactly one real root (a repeated root). The parabola touches the x-axis at one point.
• D < 0: There are two complex (imaginary) roots. The parabola does not touch the x-axis.

Step-by-Step Example

Let's solve the equation: 2x² – 4x – 6 = 0

1. Identify coefficients: a = 2, b = -4, c = -6.
2. Calculate Discriminant: (-4)² – 4(2)(-6) = 16 + 48 = 64.
3. Apply Formula: x = [4 ± √64] / (2 * 2)
4. Solve: x = [4 ± 8] / 4.
5. Final Roots: x₁ = (12/4) = 3 and x₂ = (-4/4) = -1.

Practical Applications

Quadratic equations are used in various fields:

• Projectile Motion: Calculating the trajectory of an object thrown into the air.
• Business: Profit optimization and revenue modeling.
• Architecture: Designing curved structures and arches.
• Science: Modeling accelerated motion and gravitational pull.

function calculateQuadratic() { var a = parseFloat(document.getElementById('coeffA').value); var b = parseFloat(document.getElementById('coeffB').value); var c = parseFloat(document.getElementById('coeffC').value); var resPanel = document.getElementById('results-panel'); var errPanel = document.getElementById('error-panel'); var discSpan = document.getElementById('discVal'); var natureSpan = document.getElementById('rootNature'); var rootsSpan = document.getElementById('finalRoots'); var formulaSpan = document.getElementById('formula-display'); // Reset views resPanel.style.display = 'none'; errPanel.style.display = 'none'; if (isNaN(a) || isNaN(b) || isNaN(c)) { errPanel.innerHTML = "Error: Please enter valid numbers for all three coefficients."; errPanel.style.display = 'block'; return; } if (a === 0) { errPanel.innerHTML = "Error: Coefficient 'a' cannot be zero. If a = 0, the equation is linear, not quadratic."; errPanel.style.display = 'block'; return; } // Display current equation var eqText = a + "x² " + (b >= 0 ? "+ " + b : "- " + Math.abs(b)) + "x " + (c >= 0 ? "+ " + c : "- " + Math.abs(c)) + " = 0″; formulaSpan.innerText = "Equation: " + eqText; var discriminant = (b * b) – (4 * a * c); discSpan.innerText = discriminant.toFixed(2); if (discriminant > 0) { var x1 = (-b + Math.sqrt(discriminant)) / (2 * a); var x2 = (-b – Math.sqrt(discriminant)) / (2 * a); natureSpan.innerText = "Two Real Roots"; rootsSpan.innerHTML = "x₁ = " + x1.toFixed(4) + "x₂ = " + x2.toFixed(4); } else if (discriminant === 0) { var x = -b / (2 * a); natureSpan.innerText = "One Real Root"; rootsSpan.innerHTML = "x = " + x.toFixed(4); } else { var realPart = (-b / (2 * a)).toFixed(4); var imaginaryPart = (Math.sqrt(Math.abs(discriminant)) / (2 * a)).toFixed(4); natureSpan.innerText = "Two Complex Roots"; rootsSpan.innerHTML = "x₁ = " + realPart + " + " + imaginaryPart + "ix₂ = " + realPart + " – " + imaginaryPart + "i"; } resPanel.style.display = 'block'; }