An algebra problem by Dane Humiwat

You can convert this into the form of perfect square. $x^2+3x+6 = (x+1.5)^2+3.75$ . Given the value of $n^2 \geq 0$ for real number $n$ , there will be no real solution as the form of this equation never goes below 3.75. The answer to this equation will then be the complex number of $x= \frac{-3\pm i \sqrt{15}}{2}$

You can tell whether the roots of a trinomial are real or imaginary by finding whether the discriminant is positive or negative, respectively.

$Discriminant:\quad { b }^{ 2 }-\quad 4ac\\ a\quad =\quad 1\\ b\quad =\quad 3\\ c\quad =\quad 6\\ \\ Plugging\quad into\quad the\quad equation:\\ { 3 }^{ 2 }\quad -\quad 4(1)(6)\quad =\quad 9\quad -\quad 24\quad =\quad -15$

Because the discriminant is negative, we can conclude that the trinomial is imaginary/complex.