# Differential Calculus question: Finding the formula

The line l:

• Intersects the line $f(x) = \sqrt{x^2 +9} - x^2 + 5x$

• Is parallel to the line $m: y = 5x-2$

Find the formula of line l algebraically.

So, I found the derivative of f, which is $\dfrac{x} {\sqrt{x^2+9}} - 2x+5$, and I know that line l has a slope of 5. So I got to the point $\dfrac{x} {\sqrt{x^2+9}} - 2x = 0$, but I don't know how to solve this..

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Subtract 5 from both sides. – Stefan Oct 9 '12 at 18:13
Yes, but after that you still have something I can't solve. – JohnPhteven Oct 9 '12 at 18:16
Add $2x$ to both sides, and then square both sides. – process91 Oct 9 '12 at 18:16
To be honest, I think something is up with the question. The only line that is tangent to $m$ is $m$, so that's your answer. It would make a lot more sense if it was tangent to $f$ and intersected $m$ (which seems to be what the question was, given your work). – process91 Oct 9 '12 at 18:18
There are many lines that intersect the graph of $f$ and are parallel to the line $m$. ($f$ is not a line.) The line $m$ itself intersects $f$. – copper.hat Oct 9 '12 at 18:30

You have $\frac{x}{\sqrt{x^2+9}} - 2x = x(\frac{1}{\sqrt{x^2+9}}-2)$. So either $x = 0$ or $\frac{1}{\sqrt{x^2+9}}=2$ or $\frac{1}{x^2+9}=4$ or $4(x^2+9) = 1$. This one you should be able to solve.
Why $4(x^2+9)=1$? Where does that come from? – JohnPhteven Oct 9 '12 at 18:30