Answer:
The roots are
[tex]x=\frac{-5+i\sqrt{3}}{2}[/tex] and [tex]x=\frac{-5-i\sqrt{3}}{2}[/tex]
Step-by-step explanation:
we have
[tex]x^{2} +5x+7[/tex]
To find the roots equate the polynomial to zero and cmplete the square
[tex]x^{2} +5x+7=0[/tex]
[tex]x^{2} +5x=-7[/tex]
[tex](x^{2} +5x+2.5^{2})=-7+2.5^{2}[/tex]
[tex](x^{2} +5x+6.25)=-0.75[/tex]
rewrite as perfect squares
[tex](x+2.5)^{2}=-0.75[/tex]
take square root both sides
[tex](x+2.5)=(+/-)\sqrt{-\frac{3}{4}}[/tex]
Remember that
[tex]i=\sqrt{-1}[/tex]
substitute
[tex](x+2.5)=(+/-)i\sqrt{\frac{3}{4}}[/tex]
[tex](x+\frac{5}{2})=(+/-)i\frac{\sqrt{3}}{2}[/tex]
[tex]x=-\frac{5}{2}(+/-)i\frac{\sqrt{3}}{2}[/tex]
therefore
The roots are
[tex]x=\frac{-5+i\sqrt{3}}{2}[/tex]
[tex]x=\frac{-5-i\sqrt{3}}{2}[/tex]
