A quadratic equation can have zero, one, or two real solutions.
A quadratic equation is a polynomial equation of degree two. It can be written in the general form:
ax2 + bx + c = 0
where a, b, and c are constants, and a ≠ 0. The solutions to this equation are also called its roots. The number of real solutions depends on the discriminant, which is given by:
Δ = b2 - 4ac
Here's how the discriminant determines the number of real solutions:
| Discriminant (Δ) | Number of Real Solutions | Explanation | Example |
|---|---|---|---|
| Δ > 0 | Two distinct real solutions | The equation has two different real numbers as solutions. | x2 - 5x + 6 = 0 has solutions x=2, x=3 |
| Δ = 0 | One real solution (repeated) | The equation has one real number as a solution (also called a repeated root). | x2 - 4x + 4 = 0 has solution x=2 |
| Δ < 0 | Zero real solutions | The equation has no real number solutions; both solutions are complex numbers. | x2 + 1 = 0 has no real solutions |
Examples:
-
Two Real Solutions: The quadratic equation x2 - 3x + 2 = 0 has two real solutions, x = 1 and x = 2. The discriminant is (-3)2 - 4(1)(2) = 9 - 8 = 1 > 0.
-
One Real Solution: The quadratic equation x2 - 4x + 4 = 0 has one real solution, x = 2. The discriminant is (-4)2 - 4(1)(4) = 16 - 16 = 0.
-
Zero Real Solutions: The quadratic equation x2 + x + 1 = 0 has no real solutions. The discriminant is (1)2 - 4(1)(1) = 1 - 4 = -3 < 0. The solutions are complex numbers.
In summary, a quadratic equation can have 0, 1, or 2 real solutions, determined by the value of its discriminant.
