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Thread: Equation

  1. #1
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    Equation

    a sqrt a +b sqrt b = 183 and
    a sqrt b + b sqrt a = 182

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  2. #2
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    Re: Equation

    Let x = \sqrt{a} and y = \sqrt{b}

    So a\sqrt{a} + b\sqrt{b} = 183 becomes x^3 + y^3 = 183, while
    a\sqrt{b} + b\sqrt{a} = 182 becomes x^2y + y^2x = 182.

    Use the cubic expansion
    (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 = x^3 + y^3 + 3(x^2y + y^2x)
    and plug things in. Can you take it from here?


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  3. #3
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    Re: Equation

    Pls continue if you don't mind. Thanks in advance

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  4. #4
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    Re: Equation

    Use the cubic expansion
    (x + y)^3 = x^3 + 3x^2y + 3xy^2 + y^3 = x^3 + y^3 + 3(x^2y + y^2x)
    and plug things in.

    (x + y)^3
    = x^3 + 3x^2y + 3xy^2 + y^3
    = x^3 + y^3 + 3(x^2y + y^2x)
    = 183 + 3(182)
    = 729

    So
    (x + y)^3 = 729, which means that x + y = 9. Solve this for y to get y = 9 - x.

    Plug 9 - x in for y in
    x^3 + y^3 = 183 to get
    x^3 + (9 - x)^3 = 183.
    Solve for x.

    Go back to y = 9 - x, and plug in your x-value to get y. Finally, plug into
    x = \sqrt{a} and y = \sqrt{b}
    to find a and b.

    No more hints from me! You should be able to figure the rest out.


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