Question: Solve: $4×3^{x+1} - 9^x = 27$


Solution:
Given,

$4×3^{x+1} - 9^x = 27$

$or, 4×3^x×3^1 - 9^x = 27$

$or, 12×3^x - (3^2)^x = 27$

$or, 12×3^x - (3^x)^2 = 27$

[ Let 3^x = a ]

$or, 12a - a^2 = 27$

$or, a² - 12a +27 = 0$

$or, a² - (9+3)a + 27 = 0$

$or, a² -9a -3a +27= 0$

$or, a(a-9) - 3(a-9) = 0$

$or, (a-3)(a-9) = 0$

Either,

$a-3 = 0$

$or, 3^x = 3^1$

$\therefore x= 1$

Or,

$a -9 = 0$

$or, 3^x = 9$

$or, 3^x = 3²$

$\therefore x = 2$

Hence, the possible values of x are 1 and 2.

Related Notes and Solutions:

Here is the website link to the notes of Indices.

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