# Rewriting a quadratic function in terms of the independent variable

I have a CDF defined as

$$2(x+\frac{1}{x}-2) \text{ for } 1 \leq x \leq 2$$

I need to convert this into a percentile function and I'm getting a little lost in the algebra. Here's what I've done so far.

$$F(x) = 2x + \frac{2}{x} - 4 = \eta_{p}$$

$$\implies 2x^2 -(4 + \eta_{p})x + 2 = 0$$

I think I'm supposed to use the quadratic formula

$$\frac{4 + \eta_p \pm \sqrt{(n_p+4)^2 - 16}}{4} = 0$$

but I don't see how this helps me.

• you should write $x = \frac{4+n_p \pm \sqrt{(n_p+4)^2-16}}{4}$, then check the two solutions to see which it is. One will might unfeasible. – George Dewhirst Dec 3 at 0:14