Let

t = x^1/3

Let's rewrite the equation

8x^(2/3) - 23x^(1/3)-3=0

8*(x^(1/3))^2 - 23*(x^(1/3))-3=0 

by plugging in t = x^1/3

8*t^2 - 23*t-3=0 

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t1 = 3

t2 = -1/8

Now we go back to calculating "x" from

x^(1/3) = 3

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x1 = 27 is the first solution

Second solution we get from solving

x^(1/3) = -1/8

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x2 = -1/512 is the second solution