Question

find all complex solutions of 3x^2 + 3x + 7 = 0. (if there is more than one solution, separate them with commas.) x =

Explanation:

Step1: Identify coefficients

For the quadratic equation $3x^{2}+3x + 7=0$, we have $a = 3$, $b = 3$, $c = 7$.

Step2: Calculate the discriminant

The discriminant $\Delta=b^{2}-4ac$. Substitute the values: $\Delta=(3)^{2}-4\times3\times7=9 - 84=- 75$.

Step3: Use the quadratic formula

The quadratic formula is $x=\frac{-b\pm\sqrt{\Delta}}{2a}$. Substitute $a = 3$, $b = 3$, $\Delta=-75$: $x=\frac{-3\pm\sqrt{-75}}{6}=\frac{-3\pm5\sqrt{3}i}{6}=-\frac{1}{2}\pm\frac{5\sqrt{3}}{6}i$.

Answer:

$x=-\frac{1}{2}+\frac{5\sqrt{3}}{6}i,-\frac{1}{2}-\frac{5\sqrt{3}}{6}i$