- #1
LiHJ
- 43
- 2
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