Question

solve for x: 9^x+2-6.3^x+1=0

Answer 1

Given expression is: 9^(n+2) - 6 x 3^(n+1) + 1 = 0

Rearrange the given expression as below:

9² x 9ⁿ - 6 x 3¹ x 3ⁿ + 1 = 0

or, 81 x (3 x 3)ⁿ - 6 x 3 x 3ⁿ + 1 = 0

or, 81 x 3ⁿ x 3ⁿ - 18 x 3ⁿ + 1 = 0

Let 3ⁿ = k, then the above expression is rewritten as:

81 x k x k - 18 x k + 1 = 0

or, 81k² - 18k + 1 = 0

or, 81k² - 9k - 9k  + 1 = 0

or, 9k(9k - 1) -1(9k -1) = 0

or, (9k - 1)(9k - 1) = (9k - 1)² = 0 

or, k = 1/9

or, 3ⁿ = 1/9 = (9)^-1 = (3²)^-1 = (3)^-2

or, n = - 2

Hence the value of n = - 2

Note: for simplicity I replaced x => n (Multiplication sign and variable(x) looks like same, so don't confuse with x and n)

Hope it helps you!