Question:
If $x=\frac{-1}{2}$ is a root of the quadratic equation $3 x^{2}+2 k x+3=0$, find the values of $k$.
Solution:
Since, $x=\frac{-1}{2}$ is a root of the quadratic equation $3 x^{2}+2 k x+3=0$,
then, it must satisfies the equation.
$3\left(-\frac{1}{2}\right)^{2}+2 k\left(-\frac{1}{2}\right)+3=0$
$\Rightarrow 3\left(\frac{1}{4}\right)-k+3=0$
$\Rightarrow \frac{3}{4}-k+3=0$
$\Rightarrow \frac{3-4 k+12}{4}=0$
$\Rightarrow 3-4 k+12=0$
$\Rightarrow 4 k=15$
$\Rightarrow k=\frac{15}{4}$
Hence, the value of $k$ is $\frac{15}{4}$.