Question

If a = 5+2√6 then find a²-1÷a²

Answer 1

Answer:

a²+1/a² = 7

Step-by-step explanation:

Given, a = (3+√5)/2

We have to find the value of a² + 1/a².

1/a = 1/((3+√5)/2) = 2/(3+√5)

By taking conjugate,

2/(3+√5) = 2/(3+√5) × (3-√5)/(3-√5)

= 2(3-√5) / (3+√5)(3-√5)

By using algebraic identity,

(a - b)(a + b) = a² - b²

(3+√5)(3-√5) = (3)² - (√5)²

= 9 - 5

= 4

2(3-√5) / (3+√5)(3-√5) = 2(3-√5) / 4

So, 1/a = (3-√5)/2

Now, a² = [(3+√5)/2]²

= (3+√5)²/4

By using algebraic identity,

(a + b)² = a² + 2ab + b²

(3+√5)² = (3)² + 2(3)(√5) + (√5)²

= 9 + 6√5 + 5

= 14 + 6√5

a² = (14+6√5)/4

= 2(7+3√5)/4

a² = (7+3√5)/2

Now, 1/a² = [(3-√5)/2]²

= (3-√5)²/4

By using algebraic identity,

(a - b)² = a² - 2ab + b²

(3-√5)² = (3)² - 2(3)(√5) + (√5)²

= 9 - 6√5 + 5

= 14 - 6√5

1/a² = (14-6√5)/4

1/a² = (7-3√5)/2

a²+1/a² = (7+3√5)/2 + (7-3√5)/2

= 7/2 + 3√5/2 + 7/2 - 3√5/2

= 7/2 + 7/2

= 7

Therefore, a²+1/a² = 7