Question

If a = 2-√5 / 2+√5 and b =2+√5/2-√5, find a ^2-b^2

Answer 1

Explanation-

Rationalizing the denominator of both a and b we get,

a = 2-√5 / 2+√5

= (2-√5)(2-√5) / (2-√5)(2+√5)

= (4+5+4√5)/(4-5)

= -(9+4√5)

and

b = 2+√5 / 2-√5

= (2+√5)(2+√5) / (2-√5)(2+√5)

= (4+5-4√5)/(4-5)

= -(9-4√5)

= (4√5-9)

now,

(a+b)= (-9-4√5+4√5-9) = -18

(a-b)= (-9-4√5-4√5+9) = -8√5

we know,

a^2-b^2 = (a+b)(a-b)

put the value we get,

a^2-b^2 = (-18)(-8√5) = 144√5