SOLUTION: Thank you in advance for help with this equation. x^4-8x^2+11=0

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Question 144548This question is from textbook
: Thank you in advance for help with this equation. x^4-8x^2+11=0 This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
x%5E4-8x%5E2%2B11=0 Start with the given equation


Let u=x%5E2. So u%5E2=x%5E4


u%5E2-8u%2B11=0 Replace x%5E4 with u%5E2. Replace x%5E2 with u


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%5E2-8%2Au%2B11=0 ( notice a=1, b=-8, and c=11)




u+=+%28--8+%2B-+sqrt%28+%28-8%29%5E2-4%2A1%2A11+%29%29%2F%282%2A1%29 Plug in a=1, b=-8, and c=11



u+=+%288+%2B-+sqrt%28+%28-8%29%5E2-4%2A1%2A11+%29%29%2F%282%2A1%29 Negate -8 to get 8



u+=+%288+%2B-+sqrt%28+64-4%2A1%2A11+%29%29%2F%282%2A1%29 Square -8 to get 64 (note: remember when you square -8, you must square the negative as well. This is because %28-8%29%5E2=-8%2A-8=64.)



u+=+%288+%2B-+sqrt%28+64%2B-44+%29%29%2F%282%2A1%29 Multiply -4%2A11%2A1 to get -44



u+=+%288+%2B-+sqrt%28+20+%29%29%2F%282%2A1%29 Combine like terms in the radicand (everything under the square root)



u+=+%288+%2B-+2%2Asqrt%285%29%29%2F%282%2A1%29 Simplify the square root (note: If you need help with simplifying the square root, check out this solver)



u+=+%288+%2B-+2%2Asqrt%285%29%29%2F2 Multiply 2 and 1 to get 2

So now the expression breaks down into two parts

u+=+%288+%2B+2%2Asqrt%285%29%29%2F2 or u+=+%288+-+2%2Asqrt%285%29%29%2F2


Now break up the fraction


u=%2B8%2F2%2B2%2Asqrt%285%29%2F2 or u=%2B8%2F2-2%2Asqrt%285%29%2F2


Simplify


u=4%2Bsqrt%285%29 or u=4-sqrt%285%29



But remember, we let u=x%5E2. So



x%5E2=4%2Bsqrt%285%29 or x%5E2=4-sqrt%285%29



Take the square root of both sides for each case

x=0%2B-sqrt%284%2Bsqrt%285%29%29 or x=0%2B-sqrt%284-sqrt%285%29%29


Expand

x=sqrt%284%2Bsqrt%285%29%29, x=-sqrt%284%2Bsqrt%285%29%29, x=sqrt%284-sqrt%285%29%29, or x=-sqrt%284-sqrt%285%29%29



So our answers are:

x=sqrt%284%2Bsqrt%285%29%29, x=-sqrt%284%2Bsqrt%285%29%29, x=sqrt%284-sqrt%285%29%29, or x=-sqrt%284-sqrt%285%29%29


which approximate to

x=+2.497212041, x=-2.497212041, x=+1.328131026, or x=+-1.328131026