MP Board · Class 10 · Mathematics · Real NumbersProve that 3 + 2√5 is an irrational number, given that √5 is irrational.

Let us assume to the contrary that 3 + 2√5 is a rational number. Then, we can find coprime integers a and b (with b ≠ 0) such that 3 + 2√5 = a/b. Rearranging the equation to isolate the irrational part, we get 2√5 = (a/b) - 3, which simplifies to 2√5 = (a - 3b)/b. Dividing both sides by 2 gives √5 = (a - 3b)/2b. Since a and b are integers, the expression (a - 3b)/2b is a rational number, which implies that √5 must be a rational number. However, this contradicts the given fact that √5 is an irrational number. Therefore, our assumption is incorrect, and 3 + 2√5 is an irrational number.