I've been given a trinomial in the form $$ax^2 + bx -c$$ and asked to find the inverse.
I'm wondering if this can be done with basic algebra such in the case:
$$5x^2 + 3x - 5$$
$$x = 5y^2 + 3y - 5 $$
$$x -5 = 5y^2 + 3y $$
$$\frac{x-5}{3} = 5y^2 + y $$
$$\frac{x-5}{\frac{3}{5}} = y^2 + y $$
$$\sqrt{\frac{x-5}{8}} = 2y $$
$$\frac{\sqrt{\frac{x-5}{8}}}{2} = y^-1$$
Or does it need to be done with a specific procedure?
What you propose to do can be achieved by rewriting the original equation as (for example)
$$ 5y^2+3y-(5+x) = 0 $$
and then solving the quadratic equation, most easily by using the quadratic formula to obtain
$$ y = \frac{-3 \pm \sqrt{20x+109}}{10} $$
Your derivation has numerous algebraic errors in it. If you subtract something from one side, you don't add it to the other; you subtract it from the other side also. If you divide one side by something, you must divide the entire other side by the same thing. And so on.