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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- Jan 8th 2008, 07:38 PMchibiusagiintersections
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

- Jan 8th 2008, 07:46 PMJhevon
- Jan 8th 2008, 07:50 PMtopsquark

When this crosses the x-axis we have y = 0. So to find these x values:

So either

or

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.

So at x = 2 the slope of the tangent to the curve is .

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

So the tangent line at (2, 0) is .

-Dan