Solve for $x$

Algebra Level 3

Find the sum of all real values of $x$ for which the following holds:

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

The answer is 0.

2 solutions

Otto Bretscher
Apr 15, 2016

The LHS is an even function (divide numerator and denominator by $216^x$ ), so that the sum of the solutions is $\boxed{0}$

WoAHHHH! This is the your best one-line solution!!!

Pi Han Goh - 5 years, 1 month ago

Amazing solution Sir!!!!!! LHS is an even function shows if $x$ is a solution, then $-x$ is also a solution to this equation, and hence, the sum of roots is always going to be $0$ . No need to solve the equation. Hats off to this one liner solution.. and I know you are accustomed to writing such solutions. If it had been possible, I would have upvoted it more than 10 times.. :D. Really liked the solution a lot.... (+1)

Raushan Sharma - 5 years, 2 months ago

This indeed is a wonderful solution and I did it the same way. A nice observation is that 8 27=12 18.. which enables us to think of the next step ... that is to divide it by 216.

@Raushan Sharma

If you upvoted the solution an even number of times, that would be equivalent to not upvoting it at all.

I saw a joke on this.... A boy tells his friend, 'your profile pic on fb is really awesome bro.' to which his friend replies, ' Really! Thanks mate.' ... the first one replies back with.. 'Yes I liked it so much I pressed the like button 2 times'.

Soumava Pal - 5 years, 2 months ago

By that $827 = 1218$ , you mean $8 \times 27 = 12 \times 18$ right?? Exactly... that observation only inspires one to divide by $216$ , and that's a great way in which you solved the problem!!

I said, I could have upvoted "more than" 10 times. Though, I know, I should have mentioned that I would upvote an odd number of times. :p BTW, nice joke...... Soumava Pal

Raushan Sharma - 5 years, 1 month ago

this question appeared in NMTC(junior level) 2015

Ayush G Rai - 5 years ago

Raushan Sharma
Apr 15, 2016

Set $p=2^x$ and $q=3^x$ . Then, we have the equation:

$\frac{p^3 + q^3}{pq(p+q)} = \frac{7}{6}$

$\Rightarrow \frac{p^2 -pq + q^2}{pq} = \frac{7}{6}$

$\Rightarrow \frac{p^2 + q^2}{pq} = \frac{13}{6}$

$\Rightarrow 6p^2 - 13pq + 6q^2 = 0$

$\Rightarrow (3p-2q)(2p-3q) = 0$

$\Rightarrow \frac{p}{q} = \frac{2}{3}, \frac{3}{2}$

Then, we get, $x = 1, -1$

So, sum of all possible real values of $x = 0$

p²-pq+q²/pq =7/6 In this when we move -pq to the RHS then it will be 7/6 +6 /\ x and when we put the second value of p to the equation then it will be p=2/\x 3=2/\x then how can it be -1 can you explain it ?

Sonia Gupta - 5 years, 2 months ago

Did exactly same way!!! :-)

Atul Shivam - 5 years, 2 months ago

Solve for x x x

The answer is 0.

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