Calculator Scientific Online

Quadratic Equation Solver

Use this calculator to find the roots (solutions) of a quadratic equation in the standard form: ax² + bx + c = 0.

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 "roots" or "solutions" of a quadratic equation are the values of x that satisfy the equation, making the entire expression equal to zero. Graphically, these are the x-intercepts where the parabola (the graph of a quadratic function) crosses the x-axis.

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

The term inside the square root, (b² - 4ac), is called the discriminant, often denoted by Δ (Delta). The value of the discriminant determines the nature of the roots:

• If Δ > 0: There are two distinct real roots. The parabola intersects the x-axis at two different points.
• If Δ = 0: 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: There are two complex conjugate roots. The parabola does not intersect the x-axis.

How to Use This Calculator

1. Identify Coefficients: Look at your quadratic equation and identify the values for a, b, and c. Remember to include their signs (positive or negative).
2. Enter Values: Input these values into the respective fields: "Coefficient 'a'", "Coefficient 'b'", and "Coefficient 'c'".
3. Calculate: Click the "Calculate Roots" button.
4. Interpret Results: The calculator will display the roots of your equation, indicating if they are real or complex.

Examples

Let's look at some common scenarios:

Example 1: Two Distinct Real Roots

Equation: x² - 3x + 2 = 0

• a = 1
• b = -3
• c = 2

Using the calculator with these values will yield: Root 1: 2, Root 2: 1

Example 2: One Real Root (Repeated)

Equation: x² - 4x + 4 = 0

• a = 1
• b = -4
• c = 4

Using the calculator with these values will yield: Root 1: 2, Root 2: 2

Example 3: Two Complex Conjugate Roots

Equation: x² + 2x + 5 = 0

• a = 1
• b = 2
• c = 5

Using the calculator with these values will yield: Root 1: -1 + 2i, Root 2: -1 – 2i

Example 4: Missing Terms

Equation: 2x² - 8 = 0 (Here, b = 0)

• a = 2
• b = 0
• c = -8

Using the calculator with these values will yield: Root 1: 2, Root 2: -2