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Prove that (4 - 5`Sqrt(2)` ) is Irrational. - Mathematics

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Prove that (4 - 5`sqrt(2)` ) is irrational.

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Solution

Let x = 4 - 5`sqrt(2)` be a rational number.
x = 4 - 5`sqrt(2)`
⇒`x^2 = (4 - 5sqrt(2)`  )2
⇒ `x^2 = 4^2 + (5sqrt(2) ) 2 – 2(4) (5 sqrt(2) )`
⇒ `x^2 = 16 + 50 – 40sqrt(2)`  
⇒ `x^2 – 66 = – 40 sqrt(2)`
⇒`( 66− x^2)/40 =sqrt(2)`
Since x is a rational number, `x^2` is also a rational number.
⇒ 66 -`x^2` is a rational number
⇒ `(66− x^2)/40` is a rational number
⇒`sqrt(2)` is a rational number
But `sqrt(2)` is an irrational number, which is a contradiction.
Hence, our assumption is wrong.
Thus, (4 - 5`sqrt(2)` ) is an irrational number.

Concept: Concept of Irrational Numbers
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