Quadratic Inequality: Solve for 4/m-2/n Without a Calculator

[LiHJ]

Homework Statement

Dear Mentors and PF helpers,

Here's the question:

The roots of a quadratic equation $$3x^2-\sqrt{24}x-2=0$$ are m and n where m > n . Without using a calculator,

a) show that $$1/m+1/n=-\sqrt{6}$$
b) find the value of 4/m - 2/n in the form $$\sqrt{a}-\sqrt{b}$$

Homework Equations

Sum of roots: m+ n= $$\sqrt{24}/3=2\sqrt{6}/3$$
Product of roots = -2/3

The Attempt at a Solution

For a):
I was able to show it:
$$1/m+1/n= (n +m)/mn$$

For b):
My method seem to be quite long, I did simultaneous equations to solve for m and n. Using the quadratic formula. There are 2 answers for both m and n. So I choose the set of m and n that fits the criteria.
$$m=(\sqrt{6}+\sqrt{12})/3$$
$$n=(\sqrt{6}-\sqrt{12})/3$$

Therefore 4/m -2/n = $$3\sqrt{12}-\sqrt{6}$$

My answers are correct but I wonder is there a shorter way to do part (b)

Thanks for your time

 

[mfb]

(a) and (b) together allow to calculate 1/m and 1/n in an easy way. No matter which approach you use, it is at most one step away from finding m and n.
You could find solutions of 1/x, that might save one or two steps, but I don't see a solution that avoids solving a quadratic equation.