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Thread: cubics

  1. #1
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    cubics

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

    $\displaystyle y=x^3-x$, $\displaystyle y=ax+a$
    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($\displaystyle x_1,y_1$) and Q$\displaystyle (x_2,y_2$), where $\displaystyle x_1<x_2$.
    1. Show that $\displaystyle x_1+x_2=1$
    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; Jan 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.

    $\displaystyle y=x^3-x$, $\displaystyle y=ax+a$
    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($\displaystyle x_1,y_1$) and Q$\displaystyle (x_2,y_2$), where $\displaystyle x_1<x_2$.
    1. Show that $\displaystyle x_1+x_2=1$
    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
    $\displaystyle y = ax + a$ meets $\displaystyle y = x^3 -x$ where

    $\displaystyle x^3 - x(a+1) - a = 0$

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

    Factorise:

    $\displaystyle (x+1)(x^2 - x - a) = 0$

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

    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 = $\displaystyle -\frac{b}{a}$" and "For question (2), say that they must be equal if the line is tangent; so $\displaystyle b^2= 4ac$", he is referring to the "a, b, c" in the general quadratic, $\displaystyle ax^2+ bx+ c$, not the "a" in this particular equation, y= ax+ a.
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