Question

Someone please help me solve this.
x^4 - 22x^3 + 159x^2 +418x - 600=0

Answer 1

does it have to be exact values?

Answer 2

Yeah ..

Answer 3

I do not know how to solve this easily without hit and trial.
It does not have 'exact values', the solutions are real and imaginary.

https://www.wolframalpha.com/input/?i=x%5E4+-+22x%5E3+%2B+159x%5E2+%2B418x+-+600%3D0

Answer 4

Do you use the factor therom and then use long division, and you should get a cubic equation then solve remeberig it equals zero?

Answer 5

The real solutions in exact form look pretty `crazy`.
The rational root theorem allows us to find `integer roots`.
But this function doesn't have integer roots, so that won't help. :(

Here is what the function looks like, if you were curious.
https://www.desmos.com/calculator/j4aiuvwttq

Answer 6

I tried to divide it with x^2 each, turning it into:
\[x^{2} - 22x + 159 + \frac{ 418 }{ x } - \frac{ 600 }{ x^{2}} = 0\]
\[\left( x^{2} - \frac{ 600 }{ x^{2} } \right) - 22\left( x - \frac{ 19 }{ x } \right) + 159 = 0\]
Thought I could change the \[\left( x - \frac{19}{x}\right) = Y\] which then makes \[\left( x^{2} - \frac{ 600 }{ x^{2} } \right) = Y^{2} ± (something)\] like the previous questions on the same type of equations but it doesn't work.

I would like to know the solution step by step, but since there really isn't an exact value, guess I'll just pass this one. Anyway, thank you for the answer, guys!