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    intersections

    Prove that the curve y=(x-2)(x^2+2x+6) crosses the x axis at on point only and find the equation of the tengent at that point
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    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by chibiusagi View Post
    Prove that the curve y=(x-2)(x^2+2x+6) crosses the x axis at on point only and find the equation of the tengent at that point
    we can find how many zeros the function has by simply solving for them.

    set $\displaystyle (x - 2) \left( x^2 + 2x + 6 \right) = 0$

    $\displaystyle \Rightarrow x - 2 = 0$ or $\displaystyle x^2 + 2x + 6 = 0$

    $\displaystyle \Rightarrow x = 2$ or $\displaystyle x = \frac {-2 \pm \sqrt{-20}}2$

    clearly the quadratic has no real roots, thus the only root is $\displaystyle x = 2$

    to find the tangent line, use the point-slope form.

    $\displaystyle y - y_1 = m(x - x_1)$

    here, $\displaystyle m = f'(2)$ and $\displaystyle (x_1,y_1) = (2,0)$
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    Forum Admin topsquark's Avatar
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    Quote Originally Posted by chibiusagi View Post
    Prove that the curve y=(x-2)(x^2+2x+6) crosses the x axis at on point only and find the equation of the tengent at that point
    $\displaystyle y = (x - 2)(x^2 + 2x + 6)$

    When this crosses the x-axis we have y = 0. So to find these x values:
    $\displaystyle (x - 2)(x^2 + 2x + 6) = 0$

    So either
    $\displaystyle x - 2 = 0 \implies x = 2$
    or
    $\displaystyle x^2 + 2x + 6 = 0$

    Note, however, that the quadratic factor here has no real zeros. So no real zeros are obtained by solving this equation.

    Thus the curve only crosses the x-axis once.

    Now to find the equation of the tangent.

    $\displaystyle y^{\prime}(x) = 3x^2 + 2$

    So at x = 2 the slope of the tangent to the curve is $\displaystyle y^{\prime}(2) = 14$.

    So we need the equation of a line with a slope of 14 that passes through the point (2, 0).

    $\displaystyle y = 14x + b$

    $\displaystyle 0 = 14 \cdot 2 + b \implies b = -28$

    So the tangent line at (2, 0) is $\displaystyle y = 14x - 28$.

    -Dan
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