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 

For the following two (02) items:
Consider the equation abx² + bcx + ca = cax² + abx + bc

If the roots of the equation are equal, then which one of the following is correct?

For the following two (02) items:
Consider the equation abx² + bcx + ca = cax² + abx + bc

If the roots of the equation are equal, then a, b, c are in