The solutions are:
[tex]x =3+2\sqrt{3}i\\\\\:x=3-2\sqrt{3}i[/tex]
Solution:
[tex]f(x) = x^2 - 6x + 21[/tex]
We have to find solutions when f(x) = 0
[tex]x^2 - 6x + 21 = 0[/tex]
Solve by quadratic formula
[tex]\mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:}\\\\x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\\\\\mathrm{For\:}\quad a=1,\:b=-6,\:c=21\\\\x=\frac{-\left(-6\right)\pm \sqrt{\left(-6\right)^2-4\cdot \:1\cdot \:21}}{2\cdot \:1}[/tex]
[tex]x = \frac{6 \pm \sqrt{\left(-6\right)^2-4\cdot \:1\cdot \:21}}{2\cdot \:1}\\\\Simplify\\\\x = \frac{6 \pm \sqrt{36 - 84}}{2}\\\\x = \frac{6 \pm \sqrt{-48}}{2}\\\\x = \frac{6 \pm i \sqrt{48}}{2}\\\\simplify\\\\x = \frac{6 \pm 4\sqrt{3}i}{2}\\[/tex]
[tex]x = 3 \pm 2 \sqrt{3} i[/tex]
We have two solutions:
[tex]x = 3 + 2 \sqrt{3} i\\\\x = 3 - 2 \sqrt{3} i[/tex]
Thus the solutions are:
[tex]x =3+2\sqrt{3}i\\\\\:x=3-2\sqrt{3}i[/tex]
