Answer:
[tex]x=\dfrac{-3+5\sqrt{3}i}{2}[/tex] and [tex]x=\dfrac{-3- 5\sqrt{3}i}{2}[/tex].
Step-by-step explanation:
The given equation is
[tex]x^2+3x+21=0[/tex]
To make it complete square add and subtract square of half of coefficient of x.
[tex]x^2+3x+21+(\dfrac{3}{2})^2-(\dfrac{3}{2})^2=0[/tex]
[tex]x^2+3x+(\dfrac{3}{2})^2+21-\dfrac{9}{4}=0[/tex]
[tex](x+\dfrac{3}{2})^2+\dfrac{75}{4}=0[/tex]
[tex](x+\dfrac{3}{2})^2=-\dfrac{75}{4}[/tex]
Taking square root on both sides.
[tex]x+\dfrac{3}{2}=\pm\sqrt{-\dfrac{75}{4}}[/tex]
[tex]x+\dfrac{3}{2}=\pm\dfrac{5\sqrt{3}i}{2}}[/tex]
[tex]x+\dfrac{3}{2}=pm \dfrac{5\sqrt{3}i}{2}}[/tex]
[tex]x=-\dfrac{3}{2}pm \dfrac{5\sqrt{3}i}{2}}[/tex]
[tex]x=\dfrac{-3\pm 5\sqrt{3}i}{2}[/tex]
Therefore, the roots of the given equation are [tex]x=\dfrac{-3+5\sqrt{3}i}{2}[/tex] and [tex]x=\dfrac{-3- 5\sqrt{3}i}{2}[/tex].
