The roots of the quadratic equation $\displaystyle 16x^2 + 7x + 4 = 0$ are $\displaystyle \alpha^2$ and $\displaystyle \beta^2$. Find two distinct quadratic equations whose roots are $\displaystyle \alpha$ and $\displaystyle \beta$.

My steps:

Sum of roots, $\displaystyle \alpha^2 + \beta^2 = \frac{-7}{16}$

Product of roots, $\displaystyle \alpha^2\beta^2 = \frac{1}{4}$

Sum of new roots, $\displaystyle \alpha + \beta = ... = \sqrt{\frac{-7}{16} + 2\sqrt\frac{1}{4}} = \pm\frac{3}{4}$

Product of new roots, $\displaystyle \alpha\beta = ... = \sqrt{\frac{1}{4}} = \pm\frac{1}{2}$

But it turns out I'm partially correct, because while there is a $\displaystyle \pm$ before $\displaystyle \frac{3}{4}$, there is not supposed to be a $\displaystyle \pm$ before $\displaystyle \frac{1}{2}$.

After staring at it for a couple minutes I'm guessing that since the original equation has a positive c/a, hence my new product of roots must be positive too. Am I right?