A Nice Algebra Problem | Can you solve this ? | Math Olympiad

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On How to solve for X in this Algebra, Maths Olympiad Problem
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[@damirdukic]

I divided both the numerator and the denominator with "8^x", then I introduced the substitution "u = (3/2)^x", thus obtaining the equation "6u^3 - 7u^2 - 7u + 6 = 0". "u = -1" is the most obvious solution. After we divide this equation with "u + 1", we obtain the equation "6u^2 - 13u + 6 = 0", which has two solutions: 2/3 and 3/2. So, "u" can be 1, 2/3 or 3/2 and since u is (3/2)^x, x can be -1 or 1. (If we search the solution in the real number set.)

[@yamunadegamboda8485]

With out using the formula can get the value of x by using factors

[@key_board_x]

@ 4:55 / 11:54 → do not make cross multiply, but divide by ab (directly) (a² - ab + b²) / ab = 7/6 (a²/ab) - (ab/ab) + (b²/ab) = 7/6 (a/b) - 1 + (b/a) = 7/6 (a/b) - 1 + [1/(a/b)] = 7/6 (a/b) + [1/(a/b)] = (7/6) + 1 → let: m = (a/b) m + (1/m) = 13/6 m² + 1 = (13/6).m m² - (13/6).m + 1 = 0 Δ = (13/6)² - 4 = (169/36) - (144/36) = 25/36 = (5/6)² m = [(13/6) ± (5/6)]/2 m = (13 ± 5)/12 m = (13 + 5)/12 = 18/12 = 3/2 m = (13 - 5)/12 = 8/12 = 2/3 Recall: m = a/b = 2^(x)/3^(x) = (2/3)^(x) First case: m = 3/2 (2/3)^(x) = 3/2 (2/3)^(x) = [1/(2/3)] (2/3)^(x) = (2/3)^(- 1) → x = - 1 Second case: m = 2/3 (2/3)^(x) = 2/3 (2/3)^(x) = (2/3)^(1) → x = 1

[@user-ee7nw2rx9s]

Поделить на 8^х (1+(3/2)^(3х))/((3/2)^(х)+(3/2)^(2х))=7/6 Легко видно что а=(3/2)^х (1+а^3)/(а^2+а)=7/6 В итоге 6а^2-13а+6=0 Легко решить И никаких лишних замен и двух переменных не нужно

[@李梓豪-x2o]

求根公式应该是2a/负b2+/-根号下b方-4ac。

[@user-ee7nw2rx9s]

Вы так легко сократили на а+b, а если a+b=0, что тогда? (a/b) =( - b+sqrt (b^2-4ac)/2a Почему-то слева b= 13, a=6 Тогда справа 6/13=3/2, что в корне не верно

[@HanhTran-tl8dm]

Good solutions, thank you.

[@CharlesChen-el4ot]

（2^x + 3^x)(2^2x + 3^2x+ 2^x*3^x)/ 6^x* ( 2^x+3^x) = 7/6 6*(2^2x + 3^2x + 2^x*3^x) = 7*6^x 6*(2^2x+3^2x)= 6^x

[@robertocarniato6195]

b^-4 a c ....

[@hansgodschalk466]

Indeed an error

[@robertocarniato6195]

@@hansgodschalk466 io faccio di peggio

[@user-de9vr5yq2h]

Nice👍

[@jyothsnapakanati5366]

Good Job! 😀

[@historyworldofarjun1870]

X = 1

[@RyanLewis-Johnson-wq6xs]

(8^x+27^x)/(12^x+18^x)=7/6 x=±1 x=± 1+(Log[1.5,(32768/14348907)e^(2kπ)]+15)i k=z

[@pspandey9737]

=>>>>> 6m^2-13m+6=0 6m^2-9m-4m+6=0 3m-2)(2m-3)=0..... .(2/3)^×=2/3,,,,,=>×=1 2m=3.....m=2/3)^× =>×=-1

[@李梓豪-x2o]

有一个步骤写错了

[@user-xq8cg3gf5f]

х = 1 !!!

[@JulioParrales-tk3iz]

VIDA SERES HUMANOS TIERRA 0

[@RyanLewis-Johnson-wq6xs]

x=±1 x=± 1+(Log[1.5,(32768/14348907)e^(2π)]+15)i final answer