Question

If x=root 5 +1by root 5-1 and y=root 5-1by root 5+1,then find the value of x^2+xy+y^2

Answer 1

X=√5+1
so X2=5+1
y2=5-1
XY =5-1
so the answer is
5+1+5-1+5-1=14

Answer 2

Answer:

Step-by-step explanation:

x = (root 5 + 1) / (root 5 - 1)

= (root 5 + 1)(root 5 + 1) / (root 5 - 1)(root 5 + 1)

= (5 + 1 + 2 root 5) / (5 - 1)

= (6 + 2 root 5) / 4

= (3 + root 5) / 2

or x2 = (9 + 5 + 6 root 5) / 4

= (14 + 6 root 5) / 4

= (7 + 3 root 5) / 2

y = (root 5 - 1) / (root 5 + 1)

= (root 5 - 1)(root 5 - 1) / (root 5 + 1)(root 5 - 1)

= (5 + 1 - 2 root 5) / (5 - 1)

= (6 - 2 root 5) / 4

= (3 - root 5) / 2

or y2 = (9 + 5 - 6 root 5) / 4

= (14 - 6 root 5) / 4

= (7 - 3 root 5) / 2

xy = [(3 + root 5) / 2] * [(3 - root 5) / 2]

= (9 - 5) / 4

= 4 / 4 = 1

so, x2+ y2+ xy

= (7 + 3 root 5) / 2 + (7 - 3 root 5) / 2 + 1

= 14 / 2 + 1

= 7 + 1 = 8

------or--------------

x2+ y2+ xy = (x + y)2 - xy

= [(3 + root 5) / 2 + (3 - root 5) / 2]2 - 1

= (6 / 2)2 - 1

= 32 - 1

= 9 - 1

= 8