Solve 2 x¹⁰ - x⁵ - 4 = 0 .

Solution

We can reduce this to quadratic in form using the substitution,

u = x⁵

u ²  = x¹⁰

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          x⁵  = 1.68614       x = (1.68614)^(1/5)   = 1.11014

u =-1.18614       x⁵  = -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.