Question

If a = (√5 + 2)/(√5 – 2) and b = (√5 – 2)/( √5 + 2), then find the value of the expression a² + b² + ab?
1. 323
2. 324
3. 320
4. 321

Answer 1

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.