Question

Does anyone know the answer?

Answer 1

a = √5

b ∈ (−∞,∞)

Explanation:

(7 + 3√5)/(3 + √5) - (7 - 3√5)/(3 - √5) = a + b√5

⇔ (7 + 3√5)(3 - √5)/(3 + √5)(3 -√5) - (7 - 3√5)(3 + √5)/(3 + √5)(3 -√5) = a + b√5

⇔ (7 + 3√5)(3 - √5) - (7 - 3√5)(3 + √5)/(3 + √5)(3 -√5) = a + b√5

⇔ 4√5/4 = a + b√5

⇔ √5 = a + b√5

⇔ b = (5 - a√5)/5

Replace b with (5 - a√5)/5

⇒ a + (5 - a√5)/5 = √5

⇔ a × 5 + (5 - a√5)(5)/5 = √5 × 5

⇔ a × 5 + 5 - a√5 = 5√5

⇔ a × 5 - √5 × a = 5√5 - 5

⇔ a(5 - √5) = 5√5 - 5

⇔ a = √5