Solve for x: 2x^(2/3) + 3x^(1/3) - 2 = 0

Question

Solve for x:

`2x^(2/3) + 3x^(1/3) - 2 = 0`

Answer 1

lemme post just to add to the superb reply above by @Lammetthash

`\bf 2x^{(2)/(3)}+3x^{(1)/(3)}-2=0\implies \stackrel{\textit{notice is just }ax^2+bx+c=0}{2\left( x^{(1)/(3)} \right)^2+3\left( x^{(1)/(3)} \right)-2}=0\\\\\\\left(2x^{(1)/(3)}-1 \right)\left(x^{(1)/(3)}+2 \right)=0\\\\-------------------------------`

`\bf 2x^{(1)/(3)}-1=0\implies 2x^{(1)/(3)}=1\implies x^{(1)/(3)}=\cfrac{1}{2}\implies x=\left( \cfrac{1}{2} \right)^3\\\\\\x=\cfrac{1^3}{2^3}\implies \boxed{x=\cfrac{1}{8}}\\\\-------------------------------\\\\x^{(1)/(3)}+2=0\implies x^{(1)/(3)}=-2\implies x=(-2)^3\implies \boxed{x=-8}`

Answer 2

`2x^(2/3)+3x^(1/3)-2=(2x^(1/3)-1)(x^(1/3)+2)=0`

Then

`2x^(1/3)-1=0\implies2x^(1/3)=1\implies x^(1/3)=\frac12\implies x=\frac18`

or

`x^(1/3)+2=0\implies x^(1/3)=-2\implies x=-8`

(assuming you are solving over the real numbers)