The maximum number of real solutions a quadratic equation can have is 2.
Understanding Quadratic Equations and Their Solutions
A quadratic equation is a polynomial equation of the second degree. The general form of a quadratic equation is:
ax2 + bx + c = 0
where a, b, and c are constants, and a ≠ 0.
The solutions (also called roots or zeros) of a quadratic equation are the values of x that satisfy the equation. These solutions represent the points where the parabola defined by the equation intersects the x-axis.
Determining the Number of Real Solutions
The number of real solutions of a quadratic equation is determined by the discriminant (Δ), which is given by:
Δ = b2 - 4ac
Here's how the discriminant relates to the number of real solutions:
- Δ > 0: The equation has two distinct real solutions.
- Δ = 0: The equation has exactly one real solution (a repeated root).
- Δ < 0: The equation has no real solutions (two complex solutions).
Maximum Number of Real Solutions
According to the provided reference, a quadratic equation can have a maximum of 2 solutions. This happens when the discriminant (Δ) is greater than zero.
Example
Consider the quadratic equation x2 - 5x + 6 = 0.
Here, a = 1, b = -5, and c = 6.
The discriminant is:
Δ = (-5)2 - 4(1)(6) = 25 - 24 = 1
Since Δ > 0, the equation has two distinct real solutions. Factoring the equation gives (x - 2)(x - 3) = 0, so the solutions are x = 2 and x = 3.
