Question

find all complex solutions of $3x^2 - x + 4 = 0$. (if there is more than one solution, separate them with commas.)

Explanation:

Step1: Identify quadratic coefficients

For $ax^2+bx+c=0$, here $a=3$, $b=-1$, $c=4$

Step2: Use quadratic formula

Quadratic formula: $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Step3: Calculate discriminant

$\Delta = b^2-4ac = (-1)^2-4(3)(4)=1-48=-47$

Step4: Substitute into formula

$x=\frac{-(-1)\pm\sqrt{-47}}{2(3)}=\frac{1\pm i\sqrt{47}}{6}$

Answer:

$\frac{1+i\sqrt{47}}{6}, \frac{1-i\sqrt{47}}{6}$