Find the value of x2 + 5xy + y2.

If x = (√2 – 1)/( √2 + 1) and y = (√2 + 1)/( √2 – 1), find the value of x2 + 5xy + y2.

Solution:

Given:
x = (√2 – 1)/( √2 + 1) and y = (√2 + 1)/( √2 – 1)
x + y = [(√2 – 1)/( √2 + 1)] + [(√2 + 1)/( √2 – 1)]
By rationalizing the denominator,
= [(√2 – 1)2 + (√2 + 1)2] / [(√2)2 – 12]
= [2 + 1 – 2√2 + 2 + 1 + 2√2] / [2 – 1]
= [6] / 1
= 6
xy = [(√2 – 1)/( √2 + 1)] × [(√2 + 1)/( √2 – 1)]
= 1
We know that
x2 + 5xy + y2 = x+ y2 + 2xy + 3xy
It can be written as
= (x + y)2 + 3xy
Substituting the values
= 62 + 3 × 1
So we get
= 36 + 3
= 39

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