Germany || A Very Nice Math Olympiad Problem | Solve for the value of x

In this video, I'll be showing you step by step on how to solve this Olympiad Maths Exponential problem using a simple trick.
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[@JakeMarley-k6g]

Interestingly, Desmos thinks there is only one solution to the equation

[@SrisailamNavuluri]

x^3+x^2=x^2(x+1)=4/27=1/9(4/3)=(1/3)^2(1/3+1
x=1/3.
Lhs=4/9(1/3)=(-2/3)^2(-2/3 +1)
x=-2/3.

[@SpencersAcademy]

Excellent delivery 👏

[@JyotiradityaDwivedi-j6e]

X^3 + X^2 - 4/27 = 0 --- (1)
X = 1/3 ( by trial and error )
divide eq (1) by (x - 1/3)
get the other value of x as -2/3 and -2/3 ( twice )
so finally three values of x are 1/3 , -2/3 , -2/3
Edit - personally i think this is a better way and quicker way , let me know if im doing something wrong

[@SpencersAcademy]

Fantastic

[@CharlesChen-el4ot]

x^3 + 1 = 3/27 - x^2
(x+1)(x^2- x+1) =
1/9 * (x+1)*(x-1)
9 x^2 -9x + 9 = x - 1
9 x^2 - 10x + 10 = 0
x = 1/18 * (( 10 +-(100-40)^1/2)

[@walterwen2975]

A Very Nice Math Olympiad Problem: x³ + x² = 4/27; x =?
x³ + x² - 4/27 = 0; (x³ - 1/27) + (x² - 3/27) = [x³ - (1/3)³] + [x² - (1/3)²] = 0
Let: a = 1/3; [x³ - (1/3)³] + [x² - (1/3)²] = (x³ - a³) + (x² - a²) = 0
(x - a)(x² + ax + a²) + (x - a)(x + a) = (x - a)(x² + ax + a² + x + a) = 0
x - a = 0, x = a or x² + ax + a² + x + a = x² + 4x/3 + 4/9 = (x + 2/3)² = 0, x + 2/3 = 0
x = 1/3 or x = - 2/3; Double root
Answer check:
x = 1/3: x³ + x² = 1/27 + 1/9 = 4/27; Confirmed
x = - 2/3: - 8/27 + 4/9 = 4/27; Confirmed
Final answer:
x = 1/3 or x = - 2/3; Double root

[@SpencersAcademy]

This is fantastic 👏

[@youssefelyousfi4929]

x=(-2/3)