Math Olympiad | A Nice Square Root Problem | There are Three Solutions

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How To Solve This Math Olympiad Square Root Simplification Problem,
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[@에스피-z2g]

rtx+rt2x=x
(1) if x=0, the equation works.
So, x=0, one answer
(2) if x is not equal to 0
1+rt2=rtx
x=3+2rt2
Answer
x=0 or x=3+2rt2

[@ΘωμάςΓκλεζάκος]

sqrt(x)+sqrt(2x)=sqrt(x)+4sqrt(x)=5sqrt(x) so 5sqrt(x)=x therefore 25x=x^2. If x=0 true. If x0 then 25=x, x in R. So x=0 or x=25, x in R. Indeed, sqrt(25)+sqrt(2*25)=sqrt(25)+4*sqrt(25)=5+(4*5)=25, same with x=0!

[@07Pietruszka1957]

x3=3-2*SQRT(2) is not a correct answer. Solution: SQRT(x)*(1+SQRT(2))=x |^2 >> x*(1+2*SQRT(2)+2)=x^2 >move everything on one side> x^2-x(3+2*SQRT(2))=0 >>> x*(x-(3+2*SQRT(2))=0 solution x1=0 or (x-(3+2*SQRT(2))=0 then x2=(3+2*SQRT(2))

[@richardmayerthaler3645]

3 - 2 * SQRT2 = 0,17157 .... SQRT 0,17157 = 0,414 / SQRT (2 * 0,17157) = 0,5857 ..... 0,414 + 0,5857 = 0,17157 ???? I think solution 3 is not korrekt.
X = 0 and X = 5,828 is correct

[@kankri99]

√x + √2x = x
√x (1 + √2) = x
1 + √2 = √x
x = (1 + √2)**2
x = 1 + 2 + 2√2
x = 3 + 2√2
Why do we need 10 minutes and 100 steps to solve this?

[@07Pietruszka1957]

You are right and Your question is very relevant! Also, what is important, the solution, x3= 3 - 2√2 is incorrect.

[@stpat7614]

There are only two solutions, not three.
sqrt(x) + sqrt(2x) = x
x = sqrt(x) + sqrt(2x)
x - sqrt(x) - sqrt(2x) = sqrt(x) + sqrt(2x) - sqrt(x) - sqrt(2x)
x - sqrt(x) - sqrt(2x) = 0
sqrt(x)*sqrt(x) - sqrt(x) - sqrt(2)*sqrt(x) = 0
sqrt(x)[sqrt(x) - 1 - sqrt(2)] = 0
Suppose sqrt(x) = 0
[sqrt(x)]^2 = 0^2
x = 0
Suppose sqrt(x) =/= 0
sqrt(x) - 1 - sqrt(2) = 0
sqrt(x) - 1 - sqrt(2) + 1 + sqrt(2) = 0 + 1 + sqrt(2)
sqrt(x) = 1 + sqrt(2)
[sqrt(x)]^2 = [1 + sqrt(2)]^2
x = 1^2 + [sqrt(2)]^2 + 2*1*sqrt(2)
x = 1 + 2 + 2*sqrt(2)
x = 3 + 2*sqrt(2)
x1 = 0, x2 = 3 + 2*sqrt(2)

[@佐藤広-q2u]

Did you upload this video to serve as a "bad example"? Why didn't you enter the solution into the given equation to check if it was correct? If you square it and remove the √, an extra solution will appear. If you substitute 3-2√2 for x on the LHS of the given equation, the LHS will become 1, which does not match the RHS.