Question

If N = [{√(√5+2) + √(√5-2)}/√(√5+1)] - √(√5+1), then find the value of N. (last answer that you gave was wrong).

Answer 1

Given, P = {√(√5 + 2) + √(√5 -2)}/√(√5 + 1) - {√(√5 + 1)}

Now, {√[√5 + 2] +√[√5 – 2]}/√[√5 + 1]

= { √[√5 + 2] + √[√5 – 2] } √[√5 – 1] / (√[√5 + 1] √[√5 – 1])

= { √[(√5 + 2)(√5 – 1)] + √[(√5 – 2)(√5 – 1)] } / √[(√5 + 1)(√5 – 1)]

= { √[5 – √5 + 2√5 – 2] + √[5 – √5 – 2√5 + 2] } / √[5 – 1]

= { √[3 + √5] + √[7 – 3√5] } / 2

= { √[(6 + 2√5)/2] + √[(14 – 6√5)/2] } / 2

= { √[(1 + 5 + 2√5)/2] + √[(9 + 5 – 6√5)/2] } / 2

= { √[(1 + √5)² /2] + √[(3 – √5)² /2] } / 2

= { (1 + √5)/√2 + (3 – √5)/√2 } / 2

= { (1 + √5) + (3 – √5) } / (2√2)

= 4 / (2√2)

= √2

So from first equation

= √2 - √(√5 + 1)]