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Math Help - two questions about quadratic equation

  1. #1
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    two questions about quadratic equation

    1.
    find the values of m for which the line y=mx-3 is a tangent to the curve y=x+(1/x) and find the x-coordinate of the point at which this tangent touches the curve.

    2.
    find the value of c and d for which {x: -5<x<3} is the solution set of
    x^2 + cx < d .
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  2. #2
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    Quote Originally Posted by wintersoltice View Post
    1.
    find the values of m for which the line y=mx-3 is a tangent to the curve y=x+(1/x) and find the x-coordinate of the point at which this tangent touches the curve.
    [snip]
    Consider the tangent at (a, a + 1/a):

    \frac{dy}{dx} = 1 - \frac{1}{x^2} \Rightarrow m = 1 - \frac{1}{a^2}.

    Therefore the equation of the tangent is:

    y - \left(a + \frac{1}{a} \right) = \left(1 - \frac{1}{a^2} \right) (x - a) \Rightarrow y = \left(1 - \frac{1}{a^2} \right) x + \frac{2}{a}.

    Therefore \frac{2}{a} = -3 \Rightarrow a = - \frac{2}{3} ......
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  3. #3
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    Quote Originally Posted by wintersoltice View Post
    [snip]2.
    find the value of c and d for which {x: -5<x<3} is the solution set of
    x^2 + cx < d .
    x^2 + cx - d < 0.

    Therefore x^2 + cx - d = (x + 5)(x - 3) = x^2 + 2x - 15 .....
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  4. #4
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    thanks for your help
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