A quadratic equation has two roots.
A quadratic equation is a polynomial equation of degree two. According to the Fundamental Theorem of Algebra, a polynomial equation of degree n has exactly n roots (counting multiplicity) in the complex number system. Therefore, a quadratic equation always has two roots.
Types of Roots
These two roots can be:
- Two distinct real roots: The quadratic equation intersects the x-axis at two different points. Example: x2 - 5x + 6 = 0 has roots x = 2 and x = 3.
- One real root (a repeated root): The quadratic equation touches the x-axis at only one point. This is also known as a double root. Example: x2 - 4x + 4 = 0 has a repeated root x = 2.
- Two distinct complex roots: The quadratic equation does not intersect the x-axis. These roots are complex conjugates. Example: x2 + 1 = 0 has roots x = i and x = -i, where i is the imaginary unit (√-1).
Determining the Nature of Roots Using the Discriminant
The discriminant (Δ) of a quadratic equation in the standard form ax2 + bx + c = 0 is given by:
Δ = b2 - 4ac
The nature of the roots can be determined by the discriminant:
| Discriminant (Δ) | Nature of Roots |
|---|---|
| Δ > 0 | Two distinct real roots |
| Δ = 0 | One real root (repeated) |
| Δ < 0 | Two distinct complex roots |
Example
Consider the quadratic equation x2 + 2x + 5 = 0. Here, a = 1, b = 2, and c = 5.
The discriminant is:
Δ = 22 - 4 1 5 = 4 - 20 = -16
Since Δ < 0, the quadratic equation has two distinct complex roots.
In summary, while a quadratic equation always has two roots, those roots can be real and distinct, real and repeated, or complex conjugates.
