Mathematical Olympiad | Algebra Problem | Math Olympiad Preparation

Can you solve the given rational and exponential equation for x?

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[wtspman]

Nice problem and solution. I got through the first part of step one (factoring the coefficients) then tried to split the fraction by the common denominator, but got stuck. I was trying to eliminate 3^x7^x from the denominator, and got into a trap. Factoring the sum of cubes in the numerator was the missing piece. 👍

[echandler]

Nice problem! So patient of you to go through all the steps for students who need them.

I like to reduce the clutter early by substituting for 3^x and 7^x.

Your key insight is to focus on the LHS first, factor and then divide top by bottom, giving a quadratic.

[nyamjargalluvsannyam]

The Guru, you always inspires me.It's joy to watch!
Math is so easy with your step by step tutorial.How fantastic you are!

[KayanHerdiana]

the first time when I saw this problem, I thought the value of x was 1 😂

[lybcxds]

تمرين جميل رائع حقا . شكرا جزيلا لكم. والله يحفظكم ويرعاكم ويحميكم . تحياتنا لكم من غزة فلسطين .

[nickdyne]

As you factored the u expression, i realised the answers, which one could see as obvious if you just add the numerator and denomenator. takes 5 seconds of mental arithmatic or 5 mins of algebra - very good!

[JPTaquari]

Herr professor, here there is only one value for X, ONE
Hugs from a Brazilian who loves math!

[pietergeerkens]

Very nice!

I completed the cube in the numerator, by adding and subtracting three times the denominator from the numerator, leading to the same quadratic with roughly the same work.

[sonnyclief]

good explanation. but for instance we can do fast solution = 27+347= 370 and 63+147 = 210. so 37^x/27^x = 37/27====> x=1

[anatoliy]

Nice to meet you again! Yesterday your comments were switched off. Thank you for a nice task and a good detailed explanation as always. God bless you, sir!

[theLordsboy]

Thanks for drawing my brain along into a nice orderly solution.

I wonder, why math becomes so easy with your explanation.
Waitung for others

[rcnayak_]

Sir, can we simply your Step - 2: which reads (u-1+(1/u)) = (37/21) in another simple way for solving 'u' such as this? We can now write Step - 2 as (u+(1/u)) = (37/21)+1 = 58/21. Using the algebraical identity (a - b)^2 = (a + b )^2 - 4ab, considering u = a and (1/u) as b, now {u - 1(/u)}^2 = { u + (1/u)}^2 - 4.u. (1/u) which becomes on simplifying, = (58/21)^2 - 4= (40/21)^2. So that u - (1/u) = +/- 40/21 (have two values). Let us consider + sign of u. We have u + (1/u) = 58/21 and u - (1/u) = 40/21. Therefore from this, u = (1/2){(58/21)+ (40/21)} = (1/2).(98/21) = 49/21 = 7/3 = (3/7)^-1. Now putting the value of u we get (3/7)^x = (3/7)^-1. So that x = -1 (one result). Taking the other (negative) value of u as - (40/21), we have u + (1/u) = 58/21 and u - (1/u) = - (40/21) so that u becomes = (1/2){(58-40)/21 }= 9/21 = 3/7. Putting the value of u, (3/7)^x = (3/7). So that x = 1. Therefore x = +1 and also -1.

[adrianmoscalu]

Nice ! Solution îs manipulation 3 and 7

[Shikuesi]

My working is far, far simpler than that given. At an early stage put a = 3^x and b=7^x and rewrite the equation, factorizing both top and bottom, seeing the common factor (a+b) and confidently cancelling as you know it can't be zero because then the expression would come to 1 and it isn't. Then the numerator shows that (3x7)^x = 21 so x = 1 is the obvious choice, which makes the numerator 37. Done.

[hansschotterradler]

I didn't see the sum of cubes, so I need to brush up on this stuff.
However, x = 1 is the obvious solution since
27 + 343 = 370 and 63 + 147 = 210.

[kalyanbasak]

🙏🏻 proffessor sir sharing reply x=1, -1

[mariodistefano]

Very very intriguing way to solve it! a question: I get used to see how many solution a generic equations has, seeing the grade of the x into the equation itself. But, in this case, why there are two solutions if the original equation seems to have only a grade of 1? Why it seems hidden this time so there, and in reality, are two solutions?

[shlifejourney]

Like 341
Nice sharing sir. Your way of teaching is so easy to understand... thanks for sharing very helpful video...👍👍👍

[giuseppemalaguti]

Divido N e D per 7^3x*3^3x....risulta dopo lunghi ma semplici calcoli x=1, -1