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Math Help - cubics

  1. #1
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    cubics

    consider the equations of the cubic function and the straight line.

    y=x^3-x, y=ax+a
    The straight line crosses the cubic function at point A(-1,0). Suppose that the straight line crosses the cubic function at two further points P( x_1,y_1) and Q (x_2,y_2), where x_1<x_2.
    1. Show that x_1+x_2=1
    2. Hence find the equation of the line throught the point A tangent to the cubic function at a point distinct from the point A
    Last edited by requal; January 8th 2009 at 09:48 PM. Reason: LaTax error
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  2. #2
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    Cubic

    Hello requal
    Quote Originally Posted by requal View Post
    consider the equations of the cubic function and the straight line.

    y=x^3-x, y=ax+a
    The straight line crosses the cubic function at point A(-1,0). Suppose that the straight line crosses the cubic function at two further points P( x_1,y_1) and Q (x_2,y_2), where x_1<x_2.
    1. Show that x_1+x_2=1
    2. Hence find the equation of the line throught the point A tangent to the cubic function at a point distinct from the point A
    y = ax + a meets y = x^3 -x where

    x^3 - x(a+1) - a = 0

    x = -1 is a root of this equation, so (x+1) is a factor.

    Factorise:

    (x+1)(x^2 - x - a) = 0

    Then consider the roots of the quadratic. For question (1), say that their sum = -\frac{b}{a}. For question (2), say that they must be equal if the line is tangent; so b^2 = 4ac.

    Can you do it now?

    Grandad
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  3. #3
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    Slight warning: when Grandad says " For question (1), say that their sum = -\frac{b}{a}" and "For question (2), say that they must be equal if the line is tangent; so b^2= 4ac", he is referring to the "a, b, c" in the general quadratic, ax^2+ bx+ c, not the "a" in this particular equation, y= ax+ a.
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