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Question

Solve for $x$ : $3^{x + 1} + 3^{2 x + 1} = 270$

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Solution

$3^{x + 1} + 3^{2 x + 1} = 270.$
$\Rightarrow {3.3}^{x} + 3^{2 x} .3 = 270$
$\Rightarrow 3^{x} + 3^{2 x} = 90$

Substituting $3^{x} = a$ we get $a + a^{2} = 90$
$\Rightarrow a^{2} + a - 90 = 0 \Rightarrow a^{2} + 10 a - 9 a - 90 = 0$
$\Rightarrow (a + 10) (a - 9) = 0 \Rightarrow a = 9$ or $a = - 10$

If $3^{x} = 9$ then x = 2 If $3^{x} = - 10$ which is not possible
$∴ x = 2$

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