France | A Nice Algebra Problem | Math Olympiad

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Find the value of x?
How to solve x = √(3x + √4x)

In this video, we'll show you How to Solve Math Olympiad Question A Nice Algebra Problem x = √(3x + √4x) in a clear , fast and easy way. Whether you are a student learning basics or a professtional looking to improve your skills, this video is for you. By the end of this video, you'll have a solid understanding of how to solve math olympiad exponential equations and be able to apply these skills to a variety of problems.
#matholympiad #maths #math #algebra

[pix-starframe]

I think I have a more simple straight forward solution, less calculation:
let’s call: y = sqrt(4x)
then we have:
y^2 = 4x
x^2 = 3x + y
Subtracting these 2 equations gives:
y^2 - x^2 = x - y
Algebraic identity gives:
(y-x)(y+x) + (y-x) = 0
meaning:
(y-x)(x+y+1)=0
2 cases then:
1/ y-x = 0 => y=x => x^2=4x => x*(x-4)=0
x=0 and x=4 are solutions
2/ x+y+1 = 0 but Delta < 0 so only complex solutions.
done.

[musicsubicandcebu]

2:29: Notice the sum of the coefficients of the quartic sum to zero, so x = 1.

[에스피-zg]

x>0
x^2=3x+2rtx
rtx=t. t>0
t^4-3t^2-2t=0
t^3-3t-2=0
(t-2)(t+1)^2=0
t>0, t=2
rtx=2, x=4

[larry]

I was so glad that the quadratic had a real solution, I forgot to check it back into the radical equation...phooey