GMAT CLUB OLYMPICS: If 3^(2x+1) + 4(3^x) 20 = 0, then what is

I like to separate the constants from the variables for questions like this and rely heavily on estimation so we don't get into messy exponent calculations.

3^? + 4*3^? - 20 = 0
3^? + 4*3^? = 20

If ? = 1, we have 3+12=15
If ? = 2, we have 9+36=45

? needs to be between 1 and 2, but much closer to 1. If we are using estimation, we see a gap of 30 b/w 1 and 2, and we need to cover 5 of that 30 or approx 1/6. Since it is exponential and not linear, 1/6 will be a little bit of an overestimation, but 1 and 1/6 will be close to our answer. Also of note, the first expression hold much less weight in the overall, as it does not have 4 multiplied before the 3 base.

For our ?, we have a 2x+1 and an x. Lets look at the answer choices and do some estimation, always start in the middle, which in this case is C. Start with the second term, as the 4* multiplier at the beginning of it will drive the biggest variance in outcome.

A. 1/6
B. 1/5
C. 1/3 -> for the second term, 3^1/3 is actually pretty close to 2^1/2 (ie cube root of 3 and square root of 2, respectively). Square root of 2 is just a good one to memorize as =1.41. 1.4*4= 5.6 (14*4=56). This looks like it won't be enough. For the first term, we have 2*1/3 + 1 = 1 and 2/3. We know 3^2 is 9, which 9+5.6 <20. Since 5/3 < 2, the first term will be less than 9. Eliminate and move downwards for higher values.
D. 2/3 -> just from estimating, we can kind of already tell that this is likely to be our answer, as 1/3 got us to a little over 10, and this is doubling it while E is about tripling it, meaning E will way overshoot our target value of 20. Nonetheless, lets go through it. Starting with the second term, 3^2/3 is the same as 9^1/3 (ie cube root of 9). We know 2*2*2 = 8, so we don't need to add much to 2 to get us to 9. Lets use 2.1 for estimation purposes. 2.1*4 = 8.4. Now onto the first term. 2*(2/3)+1 = 7/3. 3^7/3 is between 9 and 27, a little less than 1/3 of the way between the two ends of the range. With a difference of 18, 1/3 of the way between them would be 15, so lets round down and call it 13 or 14. 13 or 14 + 8 is really close to 20, and we know going to E is going to give us a much bigger number. Therefore, with almost no algebra done, and very rough estimations, we know D is the best answer in well under 2 minutes.
E. 9/10

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