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Combination

Solve: 5 2 ...

Question

Solve:
$5^{2 x} = 3^{2 x} + 2 \cdot 5^{x} + 2 \cdot 3^{x}$

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Solution

$5^{2 x} = 3^{2 x} + 2 \times 5^{x} + 2 \times 3^{x} 5^{2 x} - 2 \times 5^{x} = 3^{2 x} + 2 \times 3^{x} 5^{2 x} - 2 \times 5^{x} + 1 = 3^{2 x} + 2 \times 3^{x} + 1 (5^{x})^{2} - 2 \times 5^{x} + 1^{2} = (3^{x})^{2} + 2 \times 3^{x} + 1^{2} (5^{x} - 1)^{2} = (3^{x} + 1)^{2} o r 5^{x} - 1 = 3^{x} + 1 \to (1) 5^{x} - 3^{x} = 2 c l e a r l y x = 1 o r 5^{x} - 1 = - (3^{x} + 1) \to (2) 5^{x} + 3^{x} = 0 n o t p o s s i b l e f o r a n y v a l u e s o f x$

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