Answer:
[tex]-\frac{3}{2}+\frac{\sqrt{19}i}{2}[/tex] and [tex]-\frac{3}{2}-\frac{\sqrt{19}i}{2}[/tex]
Step-by-step explanation:
[tex]3x+x^2=-7\\\Rightarrow x^2+3x+7=0[/tex]
Discriminant
[tex]D=b^2-4ac\\\Rightarrow D=3^2-4\times 7\times 1\\\Rightarrow D=-19[/tex]
If the discriminant is a negative number then the roots of the equation are imaginary
[tex]x=\frac{-b\pm \sqrtD}{2a}\\\Rightarrow x=\frac{-3\pm \sqrt{-19}}{2\times 1}\\\Rightarrow x=-\frac{3}{2}\pm \frac{\sqrt{19}i}{2}\\\Rightarrow x=-\frac{3}{2}+\frac{\sqrt{19}i}{2}, x=-\frac{3}{2}-\frac{\sqrt{19}i}{2}[/tex]
So, the two solutions are [tex]-\frac{3}{2}+\frac{\sqrt{19}i}{2}[/tex] and [tex]-\frac{3}{2}-\frac{\sqrt{19}i}{2}[/tex]
