Answer:
x = 1 - 4i or x = 1 + 4i
Step-by-step explanation:
[tex]x^2-2x+17=0\qquad\text{subtract 17 from both sides}\\\\x^2-2x=-17\\\\x^2-2(x)(1)=-17\qquad\text{add}\ 1^2\ \text{to both sides}\\\\x^2-2(x)(1)+1^2=-17+1^2\qquad\text{use}\ (a-b)^2=a^2-2ab+b^2\\\\(x-1)^2=-17+1\\\\(x-1)^2=-16<0\Rightarrow\boxed{\text{NO REAL SOLUTION}}\ because\ x^2\geq0\\\\\text{In the set of complex numbers}\\\\i=\sqrt{-1}\\\\(x-1)^2=-16\iff x-1=\pm\sqrt{-16}\\\\x-1=-\sqrt{(16)(-1)}\ \vee\ x-1=\sqrt{(16)(-1)}\\\\x-1=-\sqrt{16}\cdot\sqrt{-1}\ \vee\ x-1=\sqrt{16}\cdot\sqrt{-1)[/tex]
[tex]x-1=-4i\ \vee\ x-1=4i\qquad\text{add 1 to both sides}\\\\x=1-4i\ \vee\ x=1+4i[/tex]
