Question

determine rational numbers X and y if 4+√2/2-√2 = x-√y​

Answer 1

Given : 4+√2/2-√2 = x-√y​

To Find : rational numbers x and y

Solution:

4+√2/2-√2 = x-√y​

LHS = (4 + √2)/(2 - √2)

Multiply and divide by 2 + √2

=  (4 + √2)(2 + √2) / (2 - √2)(2 +√2)

= (8 + 4√2 + 2√2  + 2) /(4 - 2)

= (10 + 6√2)/2

= 5 + 3 √2

can not be equal to x - √y​

Correct Question can be

4+√2/2-√2 = x+√y​   or  4+√2/2+√2 = x-√y​

case 1 :

4+√2/2-√2 = x+√y​

5 + 3 √2 =  x+√y​

=> 5 + √18 =  x+√y​

x = 5  y = 18

case 2 :

4+√2/2+√2 = x-√y​

on rationalizing

 (4 + √2)(2 - √2) / (2 - √2)(2 +√2)

= (6  - 2√2)/2

= 3 - √2

x - √y​

x = 3

y = 2

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