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