Question 199251
{{{x=(x-2)*sqrt(x)}}} Start with the given equation.



{{{x^2=(x-2)^2*x}}} Square both sides.



{{{x^2=x(x-2)^2}}} Rearrange the terms.



{{{x^2=x(x^2-4x+4)}}} FOIL



{{{x^2=x^3-4x^2+4x}}} Distribute



{{{0=x^3-4x^2+4x-x^2}}} Subtract {{{x^2}}} from both sides.



{{{0=x^3-5x^2+4x}}} Combine like terms.



{{{0=x(x^2-5x+4)}}} Factor out the GCF "x"



{{{0=x(x-1)(x-4)}}} Factor



{{{x=0}}} or {{{x-1=0}}} or {{{x-4=0}}} Set each factor equal to zero



{{{x=0}}} or {{{x=1}}} or {{{x=4}}} Solve for "x" in each equation.



So the possible solutions are {{{x=0}}} or {{{x=1}}} or {{{x=4}}}



However, if you plug in {{{x=1}}}, you'll get a contradiction. So {{{x=1}}} is NOT a solution.



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Answer:


So the solutions are {{{x=0}}} or {{{x=4}}}