Solve 2 x^{10} - x^{5 }- 4 = 0 .
Solution
We can reduce this to quadratic in form using the substitution,
u = x^{5}
u ^{2} = x^{10}
By using this substitution the equation becomes,
2u ^{2 }- u - 4 = 0
It doesn't factor and thus we'll have to use the quadratic formula on it. From the quadratic formula the solutions are,
U= (1 ± √33 )/4
Now, to get back to x's we are going to require decimals values for these so,
u = (1 + √33 )/4 = 1.68614 u = (1 -√33 )/4= -1.18614
Now, using the substitution to get back to x's gives the following,
u= 1.68614 x^{5} = 1.68614 x = (1.68614)^{(1/5) }= 1.11014
u =-1.18614 x^{5} = -1.18614 x =( -1.18614)^{(1/5)} = -1.03473
Certainly we had to employ a calculator to get the last answer for these. It is one of the cause that you don't tend to see too several of these done in Algebra class. The work and/or answers tend to be a little messy.