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 :

639

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