Need help with this quadratic math equation...

Question

Question: (x+1/x)^2 - 14(x+1/x) = 72
ended up with:
x^2 + 1 - 14x^3 - 14x -70x^2 = 0

Where do I go from here? I'm trying to find a way to apply the quadratic formula to my results, but I'm having trouble. Is anyone able to help? Thank you in advance.

Answer in the book is:
-2±√3, 9±4√59

Answer 1

First, I disagree with your expansion. I get
It should start with x^4, not x^2
And as you say, that's a daunting equation to solve.

How about letting u = x+ 1/x
Then you have
u^2-14u-72 = 0
(u-18)(u+4) = 0
u = 18,-4

So now just solve
x + 1/x = 18
x^2-18x+1 = 0
I don't get the suggested answer for this one.
and
x + 1/x = -4
x^2+4x+1 = 0
This one agrees with the book.

Answer 2

Given equation: (x+1/x)² - 14(x+1/x) = 72 ...............(1)
Use the substitution
u=x+1/x .............................................................................(1a)
then (1) becomes
u² - 14u -72 = 0 .....................................(2)
Factor (2)
(u-18)(u+4)=0
solve for
u=18 or u=-4 .....................................................................(3)

Backsubstitute solutions (3) into (1a)
u=18
x+1/x=18
multiply by x (x ≠ 0)
x^2-18x+1=0 ......................................................................(4a)
Back substitute
u=-4
x^2+4x+1=0 .......................................................................(4b)

Finally, solve 4a and 4b using the quadratic formula to get appropriate answers.

Note: The book answer you posed has an extraneous "9" tagged on at the end.

Answer 3

Thanks guys, I understand how to solve this equation through this method. The only thing I don't get is why we let "u" be "x" in the first place.