Question

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Answer 1

Solution

Given :-

• x = (2 + √5)^(1/2) + (2 - √5)^(1/2)
• y = (2 + √5)^(1/2) - (2 - √5)^(1/2)

Find :-

• Value of x² + y².

Explanation

First Calculate, x².

==> x² = [(2 + √5)^(1/2) + (2 - √5)^(1/2) ]²

Or,

==> x² = [{√(2+√5)} + {√(2-√5)}]²

Using Formula

• (a + b)² = a² + b² + 2ab

==> x² = {√(2+√5)}² + {√(2-√5)}² + 2. √(2+√5).(2-√5)

==> x² = (2+√5) + (2- √5) + 2.√(2² - 5)

==> x² = (2 + 2)+(√5-√5) + 2.√(4-5)

==> x² = 4 + 2√(-1)

we know,

• √-1 = i

==> x² = 4 + 2i _____________(1)

And, Now calculate y²

==> y² = [{√(2+√5)} - {√(2-√5)}]²

==> y² = {√(2+√5)}² + {√(2-√5)}² - 2. √(2+√5).(2-√5)

==> y² = (2+√5) + (2- √5) - 2.√(2² - 5)

==> y² = (2 + 2)+(√5-√5) - 2.√(4-5)

==> y² = 4 - 2√(-1)

==> y² = 4 - 2i ______________(2)

Now, Calculate ( x² - y²)

= x² - y²

keep value by equ(1) & equ(2)

= (4 + 2i) - (4 - 2i)

= 4 + 2i - 4 + 2i

= 0 + 4i

= 4i

Or,

= √(-4) .

Hence

• Value of (x² - y²) will be = 4i = √(-4)

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