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Math Help - Two Quadratics Equations

  1. #1
    Senior Member DivideBy0's Avatar
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    Two Quadratics Equations

    Find all possible values for x and y given the equations x+y^2 = 2 and x^2 + y = 2.
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    x+(2-x^2)^2=2
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  3. #3
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    Hello, DivideBy0!

    Find all possible values for x and y given: . \begin{array}{cc}x+y^2 \:= \:2 & [1]\\ x^2 + y \:= \:2& [2]\end{array}

    From [2], we have: . y\:=\:2-x^2

    Substitute into [1]: . x + (2-x^2)^2 \;=\;2\quad\Rightarrow\quad x + 4 - 4x^2 + x^4\:=\:2

    We have the quartic: . x^4 - 4x^2 + x + 2 \:=\:0

    . . which factors: . (x - 1)(x + 2)(x^2 - x - 1)\:=\:0

    . . and has roots: . x \;=\;1,\:\text{-}2,\:\frac{1\pm\sqrt{5}}{2}

    The corresponding y-values are: . y \;=\;1,\:\text{-}2,\:\frac{1\mp\sqrt{5}}{2}


    Therefore: . (x,\,y) \;\;=\;\;(1,\,1),\;(\text{-}2,\,\text{-}2),\;\left(\frac{1+\sqrt{5}}{2},\,\frac{1-\sqrt{5}}{2}\right),\;\left(\frac{1-\sqrt{5}}{2},\,\frac{1+\sqrt{5}}{2}\right)
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