Question

Prove that \( (2+\sqrt{3})^2 \) is not a rational number.

Hint:

Zeroes of Quadratic Equation: Use the quadratic formula and verify sum-product properties.

Answer 1

Expanding:
\[ (2+\sqrt{3})^2 = 4 + 4\sqrt{3} + 3 = 7 + 4\sqrt{3} \] Since \( 4\sqrt{3} \) is irrational, \( 7 + 4\sqrt{3} \) is also irrational.
Thus, it is not a rational number.
Correct Answer: \( 7 + 4\sqrt{3} \) is not a rational number.