if X= ( 2+√ 5)^1/2 +(2-√ 5)^1/2 Y =(2+√ 5)^1/2 -(2-√5)^1/2 Then find X^2 +Y^2 find the value of a and b if x^2+y^2-xy=a+b√ 5
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x=(2+5–√)12+(2−5–√)12=2+5–√−−−−−−√+2−5–√−−−−−−√
⟹x2=2+5–√+2⋅(2+5–√)⋅(2−5–√)−−−−−−−−−−−−−−−√+2−5–√
⟹x2=4+2⋅4–5−−−√=4+2i
y=(2+5–√)12−(2−5–√)12=2+5–√−−−−−−√−2−5–√−−−−−−√
⟹y2=2+5–√−2⋅(2+5–√)⋅(2−5–√)−−−−−−−−−−−−−−−√+2−5–√
⟹y2=4−2⋅4–5−−−√=4−2i
∴ x2+y2=4+2i+4−2i=8
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