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Math Help - Simultaneous quadratic equations

  1. #1
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    Simultaneous quadratic equations

    Hi,

    I need to solve two quadratics simultaneously to find the x and y values,
    they are:

    x^2 + y^2 = 0.23
    (x-0.15)^2 + (y-0.175)^2 = 0.16

    Thank you
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  2. #2
    MHF Contributor
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    Start with the second equation and expand it out:

    (x - 0.15)^2 + (y - 0.175)^2 = 0.16

    x^2 - 0.3x + 0.15^2 + y - 0.35y + 0.175^2 = 0.16

    x^2 + y^2 - 0.3x - 0.35y = 0.16 - 0.15^2 - 0.175^2

    Now substitute in for x^2 + y^2:

    0.23 - 0.3x - 0.35y = 0.106875

    0.3x + 0.35y = 0.123125

    Now solve for y in terms of x and substitute y = f(x) back into the first equation:

    x^2 + (f(x))^2 = 0.23

    Now you have a quadratic equation in terms of x which can be solved using the quadratic formula.
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  3. #3
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    Thanks for helping me but when I substitute in the results I get from the quadratic formula into the two equations I do not get the correct result,

    Can someone show me how to get to the quadratic formula so I can check my answers please?
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  4. #4
    MHF Contributor

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    That should work. How about if YOU show your work and why you think the answer you got was wrong?
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  5. #5
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    Forming an equation in y:
    0.35y = 0.123125 - 0.3x
    y =(0.123125 - 0.3x)/0.35

    Substitute into x^2 + y^2 = 0.23 gives:
    x^2 + ((0.123125 - 0.3x)/0.35)^2 = 0.23

    Expanding brackets:
    x^2 + (0.09x^2 - 0.6x + 0.015159765)/(0.35^2) = 0.23

    Multiplying both sides by (0.35^2):
    0.1225x^2 + 0.09x^2 - 0.6x + 0.015159765 = 0.028175

    Forming quadratic equation:
    0.2125x^2 - 0.6x - 0.013015235 = 0

    Solving with quadratic formula:
    a = 0.2125
    b = -0.6
    c = -0.013015235

    b^2 = 0.36
    4ac = -0.011062949
    b^2 - 4ac = 0.371062949
    sqrt(b^2 - 4ac) = 0.609149365

    -b - ans = -0.009149365
    -b + ans = 1.209149366

    x = 2.845057332
    or
    x = -0.021527917

    Substituting back into first equation:
    when x = 2.845057332, y = NaN
    when x = -0.021527917, y = 0.4790997272900794

    Checking in equation 2:
    (-0.021527917 - 0.15)^2 + (0.4790997272900794 - 0.175)^2 = 0.12189847124672402

    The answer doesnt equal 0.16, can someone please explain what I have done wrong?
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