Question 398877: Hello. Can you please help me?
I need to solve this equation..
x^6-9x^4-x^2+9=0
Could you please show me the steps to solving this problem so that I can understand it? thanks!
Thank you!
Found 2 solutions by nerdybill, rapaljer:
Answer by nerdybill(7384)   (Show Source):
You can put this solution on YOUR website! given:
x^6-9x^4-x^2+9=0
.
group terms:
(x^6-9x^4)-(x^2-9)=0
x^4(x^2-9)-(x^2-9)=0
(x^2-9)(x^4-1)=0
But, both of the terms are "difference of squares" so:
(x-3)(x+3)(x^2-1)(x^2+1)
One more:
(x-3)(x+3)(x-1)(x+1)(x^2+1)
Answer by rapaljer(4671)  (Show Source):
You can put this solution on YOUR website! This is a classic "Factoring by Grouping" problem. You need to group the first two terms and the last two terms together, and take out common factors in each grouping. From the first two terms, you can factor out the x^4, and from the last two terms there is only a -1 to factor out.
x^6 - 9x^4 - x^2 + 9 = 0
x^4(x^2 - 9) - 1(x^2 - 9) = 0
This gives you a common factor of (x^2-9) that you can take out.
(x^2 - 9)(x^4 - 1) = 0
Each of these binomial factors factors again:
(x-3)(x+3)(x^2-1)(x^+1)=0
Factoring again:
(x-3)(x+3)(x-1)(x+1)(x^2+1)=0
Now, are you factoring with REAL solutions only, or are you expected to find complex solutions as well? For real solutions, you will have four answers:
x=3, x=-3, x=1, x=-1, and x^2+1=0 gives no real solutions. If complex solutions are allowed (or expected!), then you must also solve x^2=-1, so with complex solution, you also have x=i and x=-i.
For more factoring explanations like this, please , please see my own website. To find my website, click on my tutor name "rapaljer" anywhere in algebra.com. Click on the link there that will take you to the Homepage of my website, which is a subwebsite at Seminole State College of Florida. From my "Homepage", look for the link "MATH IN LIVING COLOR." Click on "College Algebra", and look in "Chapter 1" for "Section 1.04 Factoring." I also have some explanation on a slightly lower level in Intermediate Algebra. For this explanation, from my Homepage, look for "Basic, Intermediate, and College Algebra: One Step at a Time." Click on "Intermediate Algebra" and look in "Chapter 2" for a very thorough explanation of Factoring.
Check also for my VIDEO on factoring in Intermediate Algebra that was made a few years before I retired of me actually teaching this topic. To see the videos, from my Homepage, look for "Rapalje Videos in Living Color." Then click on "Intermediate Algebra" and look for the topic on "Factoring." If you have RealPlayer installed on your computer, then this is a FREE download. If you do not have RealPlayer, then you will have to download RealPlayer, but this too is a FREE download.
I have explanations on nearly every topic in algebra and many from geometry as well, and many videos from my younger days as well. Everything is FREE, and I don't even have advertising on the pages!! If you don't need help with math, then visit my website to see the JOKE PAGE--all clean jokes, safe for the little ears in your family!!!
Dr, Robert J. Rapalje, Retired
Seminole State College of Florida
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