Determine the nature of the roots of the following: a. 25𝑥2 + 10𝑥 + 1 = 0 b. 3𝑥2 − 12 = 0 3

Solution:
1. For equation a. 25x^2 + 10x + 1 = 0:
- Identify coefficients: a = 25, b = 10, c = 1.
- Calculate the discriminant: \Delta = b^2 - 4ac
\Delta = 10^2 - 4 \cdot 25 \cdot 1
\Delta = 100 - 100
\Delta = 0

The discriminant is 0, which means the quadratic equation has one real and repeated root.

2. For equation b. 3x^2 - 12 = 0:
- Identify coefficients: a = 3, b = 0, c = -12.
- Calculate the discriminant: \Delta = b^2 - 4ac
\Delta = 0^2 - 4 \cdot 3 \cdot (-12)
\Delta = 0 + 144
\Delta = 144

The discriminant is positive, which means the quadratic equation has two distinct real roots.