two questions about quadratic equation

**wintersoltice** · June 9th 2008, 04:36 AM

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 .

**mr fantastic** · June 9th 2008, 05:20 AM

Originally Posted by wintersoltice

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 $\frac{2}{a} = -3 \Rightarrow a = - \frac{2}{3}$ ......

**mr fantastic** · June 9th 2008, 05:23 AM

Originally Posted by wintersoltice

[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 $x^2 + cx - d = (x + 5)(x - 3) = x^2 + 2x - 15$ .....

**wintersoltice** · June 9th 2008, 05:30 AM

thanks for your help

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