Question

Find all complex solutions of 3x^2+2x+5=0 . (If there is more than one solution, separate them with commas.)

Answer 1

Answer:

$\large\boxed{-\dfrac{1}{3}-\dfrac{\sqrt{14}}{3}i,\ -\dfrac{1}{3}+\dfrac{\sqrt{14}}{3}i}$

Step-by-step explanation:

Use

$ax^2+bx+c=0\\\\x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$

$3x^2+2x+5=0\\\\a=3,\ b=2,\ c=5\\\\b^2-4ac=2^2-4(3)(5)=4-60=-56\\\\\sqrt{b^2-4ac}=\sqrt{-56}=\sqrt{(4)(-14)}=\sqrt4\cdot\sqrt{-14}=2\sqrt{-14}$

Use

$i=\sqrt{-1}$

$\sqrt{-14}=\sqrt{(-1)(14)}=\sqrt{-1}\cdot\sqrt{14}=i\sqrt{14}$

Therefore:

$x=\dfrac{-2\pm 2i\sqrt{14}}{2(3)}=-\dfrac{2}{6}\pm\dfrac{2i\sqrt{14}}{6}=-\dfrac{1}{3}\pm \dfrac{\sqrt{14}}{3}i$