Doubt by Miguel
√2+√5 = a/b
√5 = a/b - √2
Squaring both sides.
Solution :
First of all prove that √2 is irrational by contradiction method, if it is not given.
Let us assume that √2+√5 is a rational number. So it can be written in the form of a/b where a and b are co-prime positive integers.
√2+√5 = a/b
√5 = a/b - √2
Squaring both sides.
(√5)² = (a/b - √2)²
5 = (a/b)²+(√2)²-2(a/b)(√2)
5 = a²/b² + 2-2a√2/b
5-2 = a²/b²-2a√2/b
3 = a²/b²-2ab√2/b²
3 = (a²-2ab√2)/b²
3b²=a²-2ab√2
3b²-a²=-2ab√2
5 = (a/b)²+(√2)²-2(a/b)(√2)
5 = a²/b² + 2-2a√2/b
5-2 = a²/b²-2a√2/b
3 = a²/b²-2ab√2/b²
3 = (a²-2ab√2)/b²
3b²=a²-2ab√2
3b²-a²=-2ab√2
2ab√2 = -3b²+a²
2ab√2 = a²-3b²
√2 = (a²-3b²)/2ab
∵ a and b are integers.
∴ (a²-3b²)/2ab is rational.
2ab√2 = a²-3b²
√2 = (a²-3b²)/2ab
∵ a and b are integers.
∴ (a²-3b²)/2ab is rational.
⇒ √2 is also a rational number. But we know that √2 is irrational.
It means, our assumption was wrong.
Hence, √2+√5 is irrational.
It means, our assumption was wrong.
Hence, √2+√5 is irrational.