If a = 5 + 2√6 and b = 1/a, what will be the value of a² + b²?(a) 36(b) 98(c) 50(d) 80

Solution

Given,
a = 5 + 2√6
b = 1/a
= 1/(5 + 2√6)
Rationalizing the denominator, we get;
b = [1/(5 + 2√6)] [(5 – 2√6)/(5 – 2√6)]
= (5 – 2√6)/ [(5)2 – (2√6)2]
= (5 – 2√6)/ (25 – 24)
= 5 – 2√6
Now,
a² + b²= (a + b)² – 2ab
= (5 + 2√6 + 5 – 2√6)2 – 2(5 + 2√6)(5 – 2√6)
= (10)2 – 2[(5)2 – (2√6)2]
= 100 – 2(1)
= 98