Which number, when squared, equals (7+4\sqrt{3})?

(2+\sqrt{3})

Verification:


i) (2+\sqrt{3})^2 = 2^2 + (\sqrt{3})^2 + 2\cdot2\cdot\sqrt{3} = 4 + 3 + 4\sqrt{3} = 7 + 4\sqrt{3}. Hence (2+\sqrt{3}) is the required number.

Why the others fail (brief):
- (3+\sqrt{2})^2 = 9 + 2 + 6\sqrt{2} = 11 + 6\sqrt{2}, not matching the given form.
- (\sqrt{7}+2)^2 = 7 + 4 + 4\sqrt{7} = 11 + 4\sqrt{7}, contains \sqrt{7} terms.
- (\sqrt{3}+1)^2 = 3 + 1 + 2\sqrt{3} = 4 + 2\sqrt{3}, middle term 2\sqrt{3} is too small.

Exam tip: compute the middle term 2ab to match the coefficient of the surd; that quickly eliminates wrong choices.

This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Irrational numbers and real numbers.