solve for m by rewriting the equation (transposition)

Question

In the following equation how would I rewrite the equation to solve for $m$?

$$z=\frac{-4m-8+\sqrt{(4m+8)^2+4(4(mx+y-4m-4))}}{8}$$

when $x=66$ and $y=22$ and $z=10$

Answer 1

the given expression is of the form $$z=\frac{-b+\sqrt{b^2-4ac}}{2a}$$ which can be reordered to form a quadratic in$z$. $$az^2+bz+c=0$$ $a=4$,$b=4m+8$,$c=-(mx+y-4m-4)$

Answer 2

$(1)$. Multiply both sides with $8$ ; $(2)$. Add $4m+8$ to both sides. $(3)$. Square both sides.

$(4)$. Subtract $(4m+8)^2$ from both sides. $(5)$. Divide both sides through $16$. You'll get:

$$4z^2+z\,(4m+8)=m(x-4)+(y-4)$$

Now, expand the parentheses on both sides, move all terms containing m to one side, and all other terms to the other side, then divide through the factor of m, and you're done.