Question

If x = (2 + sqrt(5)) ^ (1/2) + (2 - sqrt(5)) ^ (1/2) and y = (2 + sqrt(5)) ^ (1/2) - (2 - sqrt(5)) ^ (1/2) then evaluate x ^ 2 + y ^ 2

Answer 1

Answer:

Step-by-step explanation:

We have,

$\tt{\blue{x=\sqrt{2+\sqrt{5} }+\sqrt{2-\sqrt{5} }\,\,\,\,\,and\,\,\,\,\,y=\sqrt{2+\sqrt{5} }-\sqrt{2-\sqrt{5} }}}$

Now,

$\sf{{x}^{2}+{y}^{2}}$

$\sf{=\{\sqrt{2+\sqrt{5} }+\sqrt{2-\sqrt{5} }\}^2+\{\sqrt{2+\sqrt{5} }-\sqrt{2-\sqrt{5} }\}^2}$

$\rm{\green{We\,\,know,\,\,(a+b)^2+(a-b)^2=2(a^2+b^2) }}$

So,

$\sf{x^2+y^2=2[\{\sqrt{2+\sqrt{5} }\}^2+\{\sqrt{2-\sqrt{5} }\}^2]}$

$\sf{\implies\,x^2+y^2=2[2+\sqrt{5} +2-\sqrt{5} ]}$

$\sf{\implies\,x^2+y^2=2[2 +2 ]}$

$\sf{\implies\,x^2+y^2=2\times4}$

$\sf{\implies\,x^2+y^2=8}$

Answer 2

Step-by-step explanation:

If x = (2 + sqrt(5)) ^ (1/2) + (2 - sqrt(5)) ^ (1/2) and y = (2 + sqrt(5)) ^ (1/2) - (2 - sqrt(5)) ^ (1/2) then evaluate x ^ 2 + y ^ 2