- The curve x^2 +3xy+y^2 −x−y−8=0 is defined implicitly for y. How do you solve explicitly for y as a function of x?
We can't use computer software.
Any ideas?
This curve represents a Hyperbola why you want to solve it for x....you can find all its elements and differentiate it explicitly without solving it for x...
Anyway you can form the quadratic equation x^2+(3y-1)x+(y^2-y-8)=0 and solve it. But why you want to do this you didn't tell us anything?
Because the quadratic formula says that $\displaystyle y= \dfrac{-b\pm\sqrt{b^2- 4ac}}{2a}$ satisfys the quadratic equation $\displaystyle ay^2+ by+ c= 0$ and $\displaystyle 3x- 1$ is the coefficient of y here.