SOLUTION: x4 – 3 x2 + 2 = 0. Explain how one can use the quadratic formula to solve this equation by using a change of variable. Solve this equation completely.

Algebra ->  Radicals -> SOLUTION: x4 – 3 x2 + 2 = 0. Explain how one can use the quadratic formula to solve this equation by using a change of variable. Solve this equation completely.       Log On


   

Question 122284: x4 – 3 x2 + 2 = 0.
Explain how one can use the quadratic formula to solve this equation by using a change of variable. Solve this equation completely.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Let u=x%5E2 to get

u%5E2-3u%2B2=0


Let's use the quadratic formula to solve for u:


Starting with the general quadratic

au%5E2%2Bbu%2Bc=0

the general solution using the quadratic equation is:

u+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29



So lets solve u%5E2-3%2Au%2B2=0 ( notice a=1, b=-3, and c=2)




u+=+%28--3+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29 Plug in a=1, b=-3, and c=2



u+=+%283+%2B-+sqrt%28+%28-3%29%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29 Negate -3 to get 3



u+=+%283+%2B-+sqrt%28+9-4%2A1%2A2+%29%29%2F%282%2A1%29 Square -3 to get 9 (note: remember when you square -3, you must square the negative as well. This is because %28-3%29%5E2=-3%2A-3=9.)



u+=+%283+%2B-+sqrt%28+9%2B-8+%29%29%2F%282%2A1%29 Multiply -4%2A2%2A1 to get -8



u+=+%283+%2B-+sqrt%28+1+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



u+=+%283+%2B-+1%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



u+=+%283+%2B-+1%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

u+=+%283+%2B+1%29%2F2 or u+=+%283+-+1%29%2F2

Lets look at the first part:

x=%283+%2B+1%29%2F2

u=4%2F2 Add the terms in the numerator
u=2 Divide

So one answer is
u=2



Now lets look at the second part:

x=%283+-+1%29%2F2

u=2%2F2 Subtract the terms in the numerator
u=1 Divide

So another answer is
u=1

So our solutions are:
u=2 or u=1

Now remember, we let u=x%5E2



So our solutions become



x%5E2=2 or x%5E2=1



Now take the square root of both sides for each case


x=0%2B-sqrt%282%29 or x=0%2B-sqrt%281%29



Answer:


So our answers are


x=sqrt%282%29, x=-sqrt%282%29, x=1, or x=-1