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