Solve-for-x-8-x-27-x-12-x-18-x-7-6-Mastermind-

Question Number 168247 by Mastermind last updated on 07/Apr/22

S o l v e f o r x

\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}

M a s t e r m i n d

Answered by MJS_new last updated on 07/Apr/22

\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}

\frac{(2^{x} + 3^{x}) (4^{x} - 6^{x} + 9^{x})}{(2^{x} + 3^{x}) 6^{x}} = \frac{7}{6}

6 \times 4^{x} - 13 \times 6^{x} + 6 \times 9^{x} = 0

obviously 6 \times 4 - 13 \times 6 + 6 \times 9 = 0

\Rightarrow x = 1

less obviously

\frac{6}{4} - \frac{13}{6} + \frac{6}{9} = 0

\Rightarrow x = - 1

Commented by Mastermind last updated on 07/Apr/22

T h a n k s, p l e a s e m o r e e x p l a n a t i o n

Commented by MJS_new last updated on 07/Apr/22

sorry I {can}^{'} t explain why this is obvious to me

Commented by Mastermind last updated on 07/Apr/22

O k a y,

A n y w a y s, T h a n k s s o m u c h

Answered by cortano1 last updated on 07/Apr/22

\frac{8^{x} + 27^{x}}{12^{x} + 18^{x}} = \frac{7}{6}

\Rightarrow \frac{{(2^{x})}^{3} + {(3^{x})}^{3}}{{(2^{x})}^{2} (3^{x}) + {(3^{x})}^{2} (2^{x})} = \frac{7}{6}

\Rightarrow \frac{{(\frac{2^{x}}{3^{x}})}^{3} + 1}{{(\frac{2^{x}}{3^{x}})}^{2} + (\frac{2^{x}}{3^{x}})} = \frac{7}{6}

l e t {(\frac{2}{3})}^{x} = d; d > 0

\Rightarrow \frac{d^{3} + 1}{d^{2} + d} = \frac{7}{6}

\Rightarrow \frac{(d + 1) (d^{2} - d + 1)}{d (d + 1)} = \frac{7}{6}

\Rightarrow 6 d^{2} - 6 d + 6 = 7 d

\Rightarrow 6 d^{2} - 13 d + 6 = 0

\Rightarrow (2 d - 3) (3 d - 2) = 0

\Rightarrow {\begin{cases} d = \frac{3}{2} \Rightarrow {(\frac{2}{3})}^{x} = {(\frac{2}{3})}^{- 1}; x = - 1 \\ d = \frac{2}{3} \Rightarrow {(\frac{2}{3})}^{x} = {(\frac{2}{3})}^{1}; x = 1 \end{cases}

c h e c k : x = 1 \Rightarrow \frac{8 + 27}{12 + 18} = \frac{35}{30} = \frac{7}{6}

x = - 1 \Rightarrow \frac{\frac{1}{18} + \frac{1}{27}}{\frac{1}{12} + \frac{1}{18}} = \frac{7}{6}

Commented by peter frank last updated on 07/Apr/22

good

Commented by Mastermind last updated on 07/Apr/22

y o u d i d a g r e a t J o b