For problems (19) through (23), solve the following quadratic in form equations. 19. \( x^{4}+10 x^{2}+9=0 \) 20. \( x^{\frac{2}{3}}-5 x^{\frac{1}{3}}+6=0 \) 21. \( 2 x^{\frac{1}{2}}-x^{\frac{1}{4}}=0 \) 22. \( (x+1)^{2}-8(x+1)-9=0 \) 23. \( 4(t-1)^{2}-9(t-1)=-2 \) 24. \( x^{-2}-x^{-1}-12=0 \)

To solve the equations, we'll use substitutions to transform them into quadratic forms, making them easier to handle. For problem 19, let \( y = x^2 \). The equation transforms to \( y^2 + 10y + 9 = 0 \). Using the quadratic formula, \( y = \frac{-10 \pm \sqrt{10^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1} \), we find the roots. This gives us potential values of \( y \), which we'll convert back to \( x \) by taking square roots. For problem 20, we substitute \( z = x^{\frac{1}{3}} \), obtaining \( z^2 - 5z + 6 = 0 \). Again applying the quadratic formula allows us to find \( z \), which we will cube to get values for \( x \). For problems 21 through 24, we follow similar approaches: recognizing patterns for substitution, utilizing the quadratic formula or factoring to solve for the transformed variables, and then converting back to the original variable. Clean and systematic transformations are key!