What re the roots of the equation $$\left| {{x^2} - x - 6} \right| = x + 2$$?

The equation px² + qx + r = 0 (where p, q, r all are positive) has distinct real roots a and b. Which one of the following is correct?

The number of real roots for the equation $${x^2} + 9\left| x \right| + 20 = 0$$ is

For the following two (02) items:
Let $\alpha$ and $\beta $ be the roots of the quadratic equation
$(x^{2}+(\log _{0.5}(\alpha ^{2}))x+(\log _{0.5}(\alpha ^{2}))^{4}=0$
where $a ^ 2 \ne1$ and $\log _{0.5}(\alpha ^{2})>0$ Further, $\beta ^{2}=\alpha (\log _{\alpha ^{2}}(0.5))$
 

What is the relation between α and β