Question

.If a=5+2√6 and b=1/a , then what will be the value of a2+b2 ?

Answer 1

a=5+2√6, b=1/a=1/(5+2√6)

a²+b²=a²+1/a²
=(5+2√6)²+1/(5+2√6)²
=(25+24+10√2)+1(5-2√6)²/(5+2√6)²(5-2√6)²
=(49+10√2)+(5-2√6)²/(5²-2√6²)
=(49+10√2)+(25+24-10√2)/(25-24)
=(49+10√2)+(49-10√2)/1
=49+10√2+49-10√2
=98

Answer 2

here,
a = 5 + 2√6

b = 1/a
=1/( 5 + 2√6)
=(5 -2√6)/(5+2√6)(5 -2√6)
=(5 -2√6)/{5² -(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(25 -24)
=100 -2
= 98

hence, a² + b² = 98