If $a=\frac{\sqrt{5}+2}{\sqrt{5}-2}$ and $b=\frac{\sqrt{5}-2}{\sqrt{5}+2}$, then the value of $2a^2 + 2b^2 - 5ab $ is equal to :
Answer & explanation
Correct answer: option 4
We know that,
(a + b)2 = a2 + b2 + 2ab
(a + b)(a – b) = a2 – b2
$a=\frac{\sqrt{5}+2}{\sqrt{5}-2}$
= a = [(√5 + 2)( (√5 + 2)]/[(√5 – 2)(√5 + 2)]
= a = [(√5 + 2)2]/[(√5)2 – (2)2]
= a = [5 + 4 + 4√5]/(5 – 4)
= a = (9 + 4√5)
Also,
$b=\frac{\sqrt{5}-2}{\sqrt{5}+2}$
= b = [(√5 – 2)(√5 - 2)]/[(√5 + 2)(√5 – 2)]
= b = [(√5 – 2)2]/[(√5)2 – (2)2]
= b = [5 + 4 – 4√5]/(5 – 4)
= b = (9 – 4√5)
Now, $2a^2 + 2b^2 - 5ab $
= $2((9 + 4√5))^2 + 2((9 – 4√5))^2 - 5((9 + 4√5))((9 – 4√5)) $
= 2(81 + 80 + 72√5) + 2(81 + 80 - 72√5) - 5
= 322 + 322 - 5 = 639