NMTC 2015

Algebra Level 4

Find the sum of all real values of x x such that

8 x + 27 x 1 2 x + 18 x = 7 6 \huge\ \frac { { 8 }^{ x } + { 27 }^{ x } }{ 12^{ x } + { 18 }^{ x } } = \frac { 7 }{ 6 } .


The answer is 0.

Priyanshu Mishra by Priyanshu Mishra , 20, India

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2 solutions

Nibedan Mukherjee
Jun 12, 2016

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Good solution. Up voted.

Niranjan Khanderia - 4 years, 11 months ago
Priyanshu Mishra
Dec 5, 2015

Set 2 x = a 2^{x} = a and 3 x = b 3^{x} = b in the equation and then solve.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

sir can you please elaborate it....???

Deepansh Jindal - 5 years ago

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One could also note that the function f ( x ) = 8 x + 2 7 x 1 2 x + 1 8 x f(x) = \dfrac{8^{x} + 27^{x}}{12^{x} + 18^{x}} is even, (i.e., if x x is a solution then − x -x is as well), so the sum of all solutions is 0 0 .

Brian Charlesworth - 5 years ago

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Thanks for reminding a forgotten fact.

Niranjan Khanderia - 4 years, 11 months ago

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