1.

If a = (√5 + 2)/(√5 – 2) and b = (√5 – 2)/( √5 + 2), then find the value of the expression a2 + b2 + ab?1. 3232. 3243. 3204. 321

Answer» Correct Answer - Option 1 : 323

Given:

a = (√5 + 2)/(√5 – 2)

b = (√5 – 2)/( √5 + 2)

Concept:

First rationalize the values of ‘a’ and ‘b’ and then proceed.

Formula used:

(a + b)2 = a2 + b2 + 2ab

(a + b)(a – b) = a2 – b2

Calculation:

∵ a = (√5 + 2)/(√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)

Similarly;

b = (√5 – 2)/( √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, a2 + b2 + ab = (a + b)2 – ab

⇒ (9 + 4√5 + 9 – 4√5)2 – [(9 + 4√5)( 9 – 4√5)]

⇒ (18)2 – [(9)2 – (4√5)2]

⇒ 324 – (81 – 80)

⇒ 324 – 1

⇒ 323

∴ The value of the given expression is 323.


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