Question 756600
x^3 − 7x^2 + 6x − 47 = x^2 + 1
:
x^3 - 7x^2 - x^2 + 6x - 47 - 1 = 0
:
x^3 - 8x^2 + 6x - 48 = 0
Group factor
x^2(x - 8) + 6(x - 8) = 0
Factor out (x-8)
(x - 8)(x^2 + 6) = 0
Solutions
x = 8, the only real solution
and
x^2 = -6; non-real solutions
x = +/-{{{sqrt(-6)}}}
x = + {{{i*sqrt(6)}}}
x = - {{{i*sqrt(6)}}}
:
:
Check in original problem with x=8
x^3 - 7x^2 + 6x - 47 = x^2 + 1
8^3 - 7(8^2) + 6(8) - 47 = 8^2 + 1
512 - 448 + 48 - 47 = 64 + 1
65 = 65; confirms our solution