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Rational Expression

Solve for x:3...

Question

Solve for x:
$3^{x - 2} + 3^{3 - x} = 4$

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Solution

$3^{x - 2} + 3^{3 - x} = 4 3^{x - 2} + \frac{1}{3^{x - 3}} = 4 3^{x - 2} + \frac{3}{3^{x - 2}} = 4 L e t 3^{x - 2} = t t + \frac{3}{t} = 4 t^{2} + 3 = 4 t t^{2} - 4 t + 3 = 0 t^{2} - 3 t - t + 3 = 0 t (t - 3) - 1 (t - 3) = 0 (t - 1) (t - 3) = 0 t = 1, 3 ∴ 3^{x - 2} = 1 3^{x - 2} = 3 3^{x - 2} = 3^{\circ} 3^{x - 2} = 1 x - 2 = 0 x - 2 = 1 x = 2 x = 3 ∴ x = 2, 3$

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