Question

Find the number of real solutions for the equation `(8^x + 27^x)/(12^x + 18^x) = 7/6`.

पर्याय

• 2

• 1

• 4

• Infinitely many

Answer 1

By taking 2^x = a and 3^x = b, the equation becomes `(a^3 + b^3)/(a^2b + b^2a) = 7/6`

⇒ `(a^2 + b^2 - ab)/(ab) = 7/6 ⇒ 6a^2 - 13ab + 6b^2 = 0`

⇒ (2a – 3b) (3a – 2b) = 0
∴ 2^x+1 = 3^x+1 or 2^x-1 = 3^x-1 which implies that x = 1 & x = - 1.
It is easy to check that both x = 1 & - 1 satisfy the given equation.
So, option 2 is the correct answer.