For this case we have the following equation:
[tex] (x - 5) ^ 2 + 3 (x - 5) + 9 = 0
[/tex]
We make the following change of variables:
[tex] u = x-5
[/tex]
We have then:
[tex] u ^ 2 + 3u + 9 = 0
[/tex]
Then, using the quadratic formula we have:
[tex] u = \frac{-3 +/- \sqrt{3^2 - 4(1)(9)}}{2(1)}
[/tex]
[tex] u = \frac{-3 +/- \sqrt{9 - 36}}{2} [/tex]
[tex] u = \frac{-3 +/- \sqrt{-27}}{2} [/tex]
[tex] u = \frac{-3 +/- i 3\sqrt{3}}{2} [/tex]
Solution 1:
[tex] u = \frac{-3 + i3\sqrt{3}}{2} [/tex]
Solution 2:
[tex] u = \frac{-3 - i3\sqrt{3}}{2} [/tex]
Finally, returning the change we have:
Solution 1:
[tex]x1=u1+5=\frac{-3+i3\sqrt{3}}{2}+5[/tex]
[tex] x1 = \frac{1}{2}(7+i3\sqrt{3}) [/tex]
Solution 2:
[tex]x2=u2+5=\frac{-3-i3\sqrt{3}}{2}+5[/tex]
[tex]x2 = \frac{1}{2}(7-i3\sqrt{3})[/tex]
