Question

Can you solve the equation please

Answer 1

Step-by-step explanation:

We have,

(a + b) = √5 ---- 1

(a - b) = √3 ----- 2

Adding both the equation we get,

(a + b) + (a - b) = √5 + √3

a + b + a - b = √5 + √3

2a = √5 + √3

a = (√5 + √3)/2

Thus, we get,

a = (√5 + √3)/2 ------ 3

Now,

(1/a) = 1 ÷ (√5 + √3)/2

(1/a) = 1 × 2/(√5 + √3)

(1/a) = 2/(√5 + √3)

Rationalizing the denominator,

(1/a) = [2 × (√5 - √3)]/[(√5 + √3) × (√5 - √3)]

Using the identity,

(x + y)(x - y) = x² - y²

(1/a) = 2(√5 - √3)/[(√5)² - (√3)²]

(1/a) = 2(√5 - √3)/[5 - 3]

(1/a) = 2(√5 - √3)/2

(1/a) = (2√5 - 2√3)/2 ----- 4

Now, we must find

(a + (1/a))²

From eq.3 and eq.4 we get

= [((√5 + √3)/2) + (2√5 - 2√3)/2]²

= [(√5 + √3 + 2√5 - 2√3)/2]²

= [(3√5 - √3)/2]²

Using the identity,

(a - b)² = a² - 2ab + b²

= [((3√5)² - 2(3√5)(√3) + (√3)²)/2²]

= [(3²(√5)² - 6√15 + 3)/4]

= (9(5) - 6√15 + 3)/4

= (45 - 6√15 + 3)/4

= (48 - 6√15)/4

= 2(24 - 3√15)/4

= (24 - 3√15)/2

Thus,

(a + (1/a)²) = (24 - 3√15)/2

Hence proved.

Hope it helped and believing you understood it....All the best