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Math Help - Points of intersection, circle and parabola

  1. #1
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    Points of intersection, circle and parabola

    This is the question:

    Estimate the point(s) of intersection:
    x^2 + y^2 = 9
    y = x^2 - 4x + 5

    I substituted the parabola straight into the circle eqn, like so:

    x^2 + (x^2 - 4x + 5)^2 = 9

    Expanding that out using FOIL method:

    x^2 + x^4 -4x^3 + 5x^2 -4x^3 +16x^2 -20x + 5x^2 -20x + 25 = 9

    I simplified it to:

    x^4 - 8x^3 + 27x^2 - 40x = -16

    I'm not sure how to solve for x. It says "estimate", so perhaps I'm approaching this incorrectly?
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  2. #2
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    Quote Originally Posted by Glitch View Post
    This is the question:

    Estimate the point(s) of intersection:
    x^2 + y^2 = 9
    y = x^2 - 4x + 5

    I substituted the parabola straight into the circle eqn, like so:

    x^2 + (x^2 - 4x + 5)^2 = 9

    Expanding that out using FOIL method:

    x^2 + x^4 -4x^3 + 5x^2 -4x^3 +16x^2 -20x + 5x^2 -20x + 25 = 9

    I simplified it to:

    x^4 - 8x^3 + 27x^2 - 40x = -16

    I'm not sure how to solve for x. It says "estimate", so perhaps I'm approaching this incorrectly?
    Estimate could mean anything. Perhaps you're meant to draw careful graphs and then estimate the coordinates of the intersection points. Perhaps you're meant to solve the quartic equation using a numerical procedure to get an estimate of the values of x.

    etc.
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  3. #3
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    I see. So the easiest way would be to draw it then? Out of interest, how would I solve it without estimating?
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  4. #4
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    Factor theorem and long division to factorise the equation and solve using the null factor law, if this is possible.
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  5. #5
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    Is this something a high-school student is expected to know? I don't recall learning it.
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  6. #6
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    Quote Originally Posted by Glitch View Post
    I see. So the easiest way would be to draw it then? Out of interest, how would I solve it without estimating?
    There is no easy way of solving that particular quartic. You would have to use the quartic formula (use Google) to solve it algebraically and that will be heavy weather.
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  7. #7
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    Thank you guys. I have trouble with math, and have been doing really badly at uni. I've been trying hard to start from basics so I can do better next session. I really appreciate the help.
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  8. #8
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    Quote Originally Posted by Glitch View Post
    Is this something a high-school student is expected to know? I don't recall learning it.
    Yes, in senior secondary school, the factor theorems, long division, and null factor law are all taught.

    But like Mr F says, you can't solve this particular quartic using that method. Just solve it graphically or with a CAS.
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