Qqq Calculator

Quadratic Equation Solver

Use this calculator to find the roots (solutions) of a quadratic equation in the standard form: ax² + bx + c = 0. Simply enter the coefficients a, b, and c, and the calculator will provide the real or complex roots.

Understanding Quadratic Equations

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 written as:

ax² + bx + c = 0

Where:

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

The solutions to a quadratic equation are also known as its roots. These roots represent the x-intercepts of the parabola that the quadratic equation describes when graphed.

The Quadratic Formula

The most common method to find the roots of a quadratic equation is by using the quadratic formula:

x = [-b ± √(b² - 4ac)] / 2a

This formula allows you to directly calculate the values of x once you know the coefficients a, b, and c.

The Discriminant (b² – 4ac)

The term inside the square root, (b² - 4ac), is called the discriminant. It plays a crucial role in determining the nature of the roots:

• If (b² - 4ac) > 0: There are two distinct real roots. This means the parabola intersects the x-axis at two different points.
• If (b² - 4ac) = 0: There is exactly one real root (a repeated root). The parabola touches the x-axis at exactly one point.
• If (b² - 4ac) < 0: There are two complex (non-real) roots. The parabola does not intersect the x-axis. These roots will involve the imaginary unit i, where i = √(-1).

How to Use the Calculator

1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember to include their signs (e.g., if you have x² - 3x + 2 = 0, then a=1, b=-3, c=2).
2. Enter Values: Input these values into the respective fields in the calculator.
3. Calculate: Click the "Calculate Roots" button.
4. View Results: The calculator will display the roots, indicating whether they are real or complex.

Examples

Example 1: Two Distinct Real Roots

Equation: x² - 5x + 6 = 0

• a = 1
• b = -5
• c = 6

Discriminant = (-5)² - 4(1)(6) = 25 - 24 = 1 (Positive)

Roots: x = [5 ± √1] / 2(1)

x1 = (5 + 1) / 2 = 3

x2 = (5 - 1) / 2 = 2

Example 2: One Real (Repeated) Root

Equation: x² - 4x + 4 = 0

• a = 1
• b = -4
• c = 4

Discriminant = (-4)² - 4(1)(4) = 16 - 16 = 0 (Zero)

Roots: x = [4 ± √0] / 2(1)

x = 4 / 2 = 2

Example 3: Two Complex Roots

Equation: x² + 2x + 5 = 0

• a = 1
• b = 2
• c = 5

Discriminant = (2)² - 4(1)(5) = 4 - 20 = -16 (Negative)

Roots: x = [-2 ± √-16] / 2(1)

x = [-2 ± 4i] / 2

x1 = -1 + 2i

x2 = -1 - 2i

x = " + x.toFixed(4) + "

x1 = " + x1.toFixed(4) + "

x2 = " + x2.toFixed(4) + "

x = " + x.toFixed(4) + "

x1 = " + realPart + " + " + imaginaryPart + "i

x2 = " + realPart + " – " + imaginaryPart + "i