Question 97248
Given equation
{{{ 2x^(2/3) - 5x^(1/3) - 3 =0 }}}

{{{ 2(x^(1/3))^2 - 5x^(1/3) - 3 =0 }}}



Let {{{ x^(1/3) = A }}}

the equation above transforms to 

{{{ 2A^2 - 5A -3 =0 }}}
Using the midterm factorisation we rewrite the expression as 

{{{ 2A^2 - 6A  + A -3 =0 }}}

{{{ 2A(A-3) +1(A-3) =0 }}}
{{{(A-3)(2A+1)=0}}}

therefore {{{A-3=0}}} or {{{A=3}}}
and  {{{2A +1 =0}}}  or {{{A =-1/2}}}

Substituting A back as {{{ x^(1/3)}}} 

{{{ x^(1/3)}}} = 3

Cubing both sides
{{{ x = 3^3 }}}
{{{x=27}}} is one solution

Again for A= -1/2 we have 

{{{ x^(1/3)  = -1/2}}} 
Cubing both sides 

{{{ x = -1/8}}} 

Hence x = { -1/8,27}

You can substitute these values of x in the original equation to verify the answers.