Solve equation for z.

Question

I'd like to know how to express the $z = \frac{3}{2}a$ result from the equation below... I tried to factor or to complete the square, but without success.

$3a^2=8az-4z^2$

Thanks for your help in advance.

Answer 1

$$3a^2=8az-4z^2$$ is homogeneous and can be rewritten

$$3=8\frac za-4\left(\frac za\right)^2$$ or

$$3=8w-4w^2.$$

By the classical formula,

$$w=\frac12,\frac32$$ so

$$z=\frac a2,\frac{3a}2.$$