Is that supposed to say:
-x^5 = x + 18 ??
then
x^5 + x + 18 = 0
All odd equations must have at least one real root.
A quick check will show that we have 1 real roots and 4 complex roots
www.wolframalpha.com/input/?i=y+%3D+x%5E5+%2B+x+%2B+18
with the real root around x = -2
We could use Newton's formula
x(new) = x - (x^5 + x + 18)/(5x^4 + 1)
x .... x(new)
-2 -1.8024..
-1.8024... -1.7498...
-1.7498... -1.74659399
-1.74659399 -1.746581846
-1.746581846 -1.746581846
x = -1.746581846 , matching Wolfram's answer
wow, 9 decimal accuracy after only 5 iterations
−x5=x+18
1 answer