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Math Help - Solving a Function defined implicitly.

  1. #1
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    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?
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  2. #2
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    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?
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    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.
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  4. #4
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    Re: Solving a Function defined implicitly.

    Quote Originally Posted by turbozz View Post
    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 \begin{align*} y^2 + \left( 3x - 1 \right) y + x^2 - x - 8 &= 0 \end{align*} and complete the square on the y terms.
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    Re: Solving a Function defined implicitly.

    Quote Originally Posted by Prove It View Post
    Write it as \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 y= \frac{-b\pm\sqrt{b^2- 4ac}}{2a} with a= 1, b= 3x-1, c= x^2- x- 8.
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    Re: Solving a Function defined implicitly.

    Quote Originally Posted by HallsofIvy View Post
    Or use the quadratic formula y= \frac{-b\pm\sqrt{b^2- 4ac}}{2a} with a= 1, b= 3x-1, c= x^2- x- 8.
    why is it 3x-1 ?
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    Re: Solving a Function defined implicitly.

    Because the quadratic formula says that y= \dfrac{-b\pm\sqrt{b^2- 4ac}}{2a} satisfys the quadratic equation ay^2+ by+ c= 0 and 3x- 1 is the coefficient of y here.
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