Question: Prove that 3 + 2√5 is irrational.
Explanatory Answer
Let us assume the contrary that 3 + 2√5 is rational.
So, we can express 3 + 2√5 = \\frac{a}{b}) where a and b are co-prime integers and b ≠ 0.
So, 2√5 = \\frac{a}{b}) – 3
Or, 2√5 = \\frac{a - 3b}{b})
Hence, √5 = \\frac{a - 3b}{2b})
Because a, b, 3 and 2 are integers, \\frac{a - 3b}{2b}) is rational implying that √5 is irrational.
But this contradicts with the fact that √5 is irrational.
∴ We can conclude that 3 + 2√5 is irrational.