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Math Help - What is a quadratic simultaneous equation?

  1. #1
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    What is a quadratic simultaneous equation?

    Hello all!

    I want to understand what a quadratic (or nth power) simultaneous equation (equation system) is...
    It's clear that it's not only quadratic if the part equations are quadratic. I mean: {xy=3; x+y=4} is also called quadratic, despite the parts xy and x+y are not quadratic.
    So how can you define a quadratic sim. eq.?
    Is it quadratic if the biggest number of variables multiplied with each other is two? I mean xy=3 so x, y these are two and they are multiplied, so this is quadratic. But this also doesnt seem to be fine....

    Help me out please!
    Thanks and sorry for my English!
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  2. #2
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    A quadratic equation is defined as an equation in which one or more of its terms is squared but raised to no higher power, having the general form:

    ax^2+bx+c=0 where a and b are integral coefficients, and c is a constant.

    Your two equations do not meet this criteria (I think). Although one of them xy=3 is a rectangular hyperbola. See Rectangular Hyperbolas

    The other is linear.

    The linear equation x+y=4 graphs a line that intersects the 1st quadrant branch of the hyperbola at (1, 3) and (3, 1).

    I have seen the term "Simultaneous quadratic systems" in reference to one linear equation and one quadratic equation; also, I have seen it in reference to two quadratic equations. In either case you can have more than one point of intersection (unlike simultaneous linear systems that have at most one point of intersection)

    Don't know if this helps. Maybe someone else has a perspective.
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  3. #3
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    Thanks for your reply!
    My book calls the system xy=3 x+y=4 quadratic...
    I guess because if you substitute y=4-x into xy=3 you get a quadratic equation for x.

    As I see, in terms of single equations the form of the equation does not matter. If an equation is quadratic then all other equations equivalent with it will also be quadratic.

    So maybe extending this to simultaneous equations we can define a quadratic equation system as one that can be rearranged in equvalent steps into a form where the highest degree/order* of the part-equations is 2.
    But this is just what I put together by looking at examples...

    * I mean the thing that is 2 for quadratic, 3 for cubic, I dont know the word.
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  4. #4
    A riddle wrapped in an enigma
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    Quote Originally Posted by qorilla View Post
    Thanks for your reply!
    My book calls the system xy=3 x+y=4 quadratic...
    I guess because if you substitute y=4-x into xy=3 you get a quadratic equation for x.

    As I see, in terms of single equations the form of the equation does not matter. If an equation is quadratic then all other equations equivalent with it will also be quadratic.

    So maybe extending this to simultaneous equations we can define a quadratic equation system as one that can be rearranged in equvalent steps into a form where the highest degree/order* of the part-equations is 2.
    But this is just what I put together by looking at examples...

    * I mean the thing that is 2 for quadratic, 3 for cubic, I dont know the word.
    I think you explained that very well. Good job.
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