10. \( x^{\frac{1}{2}}-3 x^{\frac{1}{4}}+2= \) 43. \( x-x^{\frac{1}{2}}-20=0 \) 16. \( x-3 x^{\frac{1}{2}}-18= \) 19. \( x^{\frac{2}{3}}-3 x^{\frac{1}{3}}-4=0 \)

For the expression \( x^{\frac{1}{2}} - 3 x^{\frac{1}{4}} + 2 \), you can simplify it by performing a substitution. Letting \( y = x^{\frac{1}{4}} \), the expression transforms to \( y^2 - 3y + 2 \), which can be factored into \( (y - 1)(y - 2) \). This leads to the solutions \( y = 1 \) or \( y = 2 \), and plugging back gives \( x = 1^4 = 1 \) or \( x = 2^4 = 16 \). Now, for \( x - x^{\frac{1}{2}} - 20 = 0 \), rearrange it as \( x - 20 = x^{\frac{1}{2}} \). Squaring both sides yields a quadratic equation \( x^2 - 40x + 400 = 0 \). Factoring gives \( (x - 20)^2 = 0 \), leading to \( x = 20 \) as a double solution, indicating it touches the x-axis at this point.