A quadratic equation has two possible solutions.
Quadratic equations, by definition, are polynomial equations of degree 2. The fundamental theorem of algebra states that a polynomial equation of degree n has n complex roots (solutions), counted with multiplicity. Therefore, a quadratic equation always has two solutions. These solutions can be:
- Two distinct real solutions: The quadratic equation intersects the x-axis at two different points.
- One real solution (a repeated root): The quadratic equation touches the x-axis at one point (the vertex of the parabola lies on the x-axis). This is often described as having two equal real solutions.
- Two distinct complex solutions (non-real): The quadratic equation does not intersect the x-axis. The solutions involve the imaginary unit i.
We can use the discriminant, denoted as Δ (Delta), of the quadratic formula to determine the nature of the roots:
- Δ = b2 - 4ac
- If Δ > 0: Two distinct real solutions.
- If Δ = 0: One real solution (repeated root).
- If Δ < 0: Two distinct complex solutions.
Example 1:
Consider the equation x2 - 5x + 6 = 0. It factors as (x - 2)(x - 3) = 0, giving solutions x = 2 and x = 3. These are two distinct real solutions.
Example 2:
Consider the equation x2 - 4x + 4 = 0. It factors as (x - 2)(x - 2) = 0, giving a solution x = 2 (repeated root). There is one distinct real solution, but it's counted as two equal solutions.
Example 3:
Consider the equation x2 + 1 = 0. The solutions are x = i and x = -i. These are two distinct complex solutions.
In summary, while the distinct number of real solutions can be one or two (or none if only real solutions are considered), the total number of solutions, considering complex and repeated roots, is always two.
