In this case, the problem is asking us to solve the equation
[tex]x^{\frac{1}{3}} - \dfrac{2}{x^{\frac{1}{3}}} = 1[/tex]
by substituting [tex]y = x^{\frac{1}{3}}[/tex].
First, let's substitute the variable into the original equation:
[tex]y - \dfrac{2}{y} = 1[/tex]
Now, let's solve:
[tex]y - \dfrac{2}{y} = 1[/tex]
[tex]y\Bigg(y - \dfrac{2}{y}\Bigg) = y(1) \Rightarrow y^2 - 2 = y[/tex]
[tex]y^2 - y - 2 = 0[/tex]
[tex](y - 2)(y + 1) = 0[/tex]
[tex]y - 2 = 0 \,\, \textrm{and} \,\, y + 1 = 0[/tex]
[tex]y = 2, -1[/tex]
Now, we aren't done yet. We have to find our answer in terms of the original variable, [tex]x[/tex]. To do this, set each value for [tex]y[/tex] into the substitution equation we found earlier, [tex]y = x^{\frac{1}{3}}[/tex]:
[tex]2 = x^{\frac{1}{3}}[/tex]
[tex]2^3 = x[/tex]
[tex]x = 8[/tex]
[tex]-1 = x^{\frac{1}{3}}[/tex]
[tex](-1)^3 = x[/tex]
[tex]x = -1[/tex]
The solutions for the equation are x = -1, 8.
