# Thread: Solving a Function defined implicitly.

1. ## Solving a Function defined implicitly.

1. The curve x^2 +3xy+y^2 xy8=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?

2. ## Re: Solving a Function defined implicitly.

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?

3. ## Re: Solving a Function defined implicitly.

Well I was told to solve this explicitly for y and then differentiate, and then use implicit differentiation in the second part of the question to demonstrate how useful this tool is.

4. ## Re: Solving a Function defined implicitly.

Originally Posted by turbozz
1. The curve x^2 +3xy+y^2 xy8=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?
Write it as \displaystyle \displaystyle \begin{align*} y^2 + \left( 3x - 1 \right) y + x^2 - x - 8 &= 0 \end{align*} and complete the square on the y terms.

5. ## Re: Solving a Function defined implicitly.

Originally Posted by Prove It
Write it as \displaystyle \displaystyle \begin{align*} y^2 + \left( 3x - 1 \right) y + x^2 - x - 8 &= 0 \end{align*} and complete the square on the y terms.
Or use the quadratic formula $\displaystyle y= \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$ with a= 1, b= 3x-1, $\displaystyle c= x^2- x- 8$.

6. ## Re: Solving a Function defined implicitly.

Originally Posted by HallsofIvy
Or use the quadratic formula $\displaystyle y= \frac{-b\pm\sqrt{b^2- 4ac}}{2a}$ with a= 1, b= 3x-1, $\displaystyle c= x^2- x- 8$.
why is it 3x-1 ?

7. ## Re: Solving a Function defined implicitly.

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.