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Homework Help: Finding Points of Intersection by Substitution

  1. Jul 19, 2010 #1
    1. The problem statement, all variables and given/known data
    Find any points of intersection of the graphs by the method of substitution.

    [tex]xy+x-2y+3=0[/tex]
    [tex]x^2+4y^2-9=0[/tex]

    2. Relevant equations


    3. The attempt at a solution

    From the second equation I can solve for y:

    [tex]y=\frac{\sqrt{9-x^2}}{2}[/tex]

    Plug it into the first equation and simplify...

    [tex]\frac{x\sqrt{9-x^2}+2x-2\sqrt{9-x^2}+6}{2}=0[/tex]

    Multiply both sides by 2 to get rid of the fraction and get the radicals over to one side, square both sides

    [tex](x\sqrt{9-x^2}-2\sqrt{9-x^2})^2=(-2x-6)^2[/tex]

    simplify and you get

    [tex]-x(x^3-4x^2-x+60)=0[/tex]

    I can find x=0 and that Is correct, but the other real root should be -3. I can graph the polynomial (x3-4x2-x+60) on my calculator and see that it has a root of -3 but how can I do it by hand? Any other way besides the rational roots test?
     
  2. jcsd
  3. Jul 19, 2010 #2

    Mark44

    Staff: Mentor

    That should be [itex]\pm[/itex].

    Also, it might be simpler to solve for one variable, say y, in the first equation.
     
  4. Jul 20, 2010 #3

    HallsofIvy

    User Avatar
    Science Advisor

    x(y+1)- 2y+ 3= 0

    x(y+1)= 2y- 3

    Solve that for x and avoid square roots.

     
  5. Jul 20, 2010 #4
    Ok, so.

    [tex]x=\frac{2y-3}{y+1}[/tex]

    You can sub that into the other equation and get

    [tex]y(4y^3-8y^2-y+6)=0[/tex] Using rational roots test you can find that the root of (4y^3-8y^2-y+6) is 3/2 and of course y= 0 as well.

    [tex]\frac{\pm1,2,3,6}{1,2,4}[/tex]

    Is this what you had in mind? It is easier than having to do the test with all the factors of 60...
     
  6. Jul 20, 2010 #5

    Mark44

    Staff: Mentor

    Yes, that's what both HallsOfIvy and I had in mind.
     
  7. Jul 20, 2010 #6

    eumyang

    User Avatar
    Homework Helper

    I'm getting -30 inside the parentheses instead of +6. And a plus 8y2 instead of a minus.


    69
     
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