Answer
The solutions are $\frac{1±\sqrt{55}}{6}$, and the solution set is {$\frac{1+\sqrt{55}}{6}, \frac{1-\sqrt{55}}{6}$}.
Work Step by Step
$\frac{x^2}{2}-\frac{x}{6}-\frac{3}{4}=0$
The least common multiple of $2$ and $6$ is $6$. Multiply the equation by the LCM.
$\frac{x^2}{2}·6-\frac{x}{6}·6-\frac{3}{4}·6=0·6$
$3x^2-x-\frac{9}{2}=0$
Divide both sides by $3$. Thus, it becomes:
$x^2-\frac{x}{3}-\frac{3}{2}=0$
Add $\frac{3}{2}$ to both sides:
$x^2-\frac{x}{3}-\frac{3}{2}+\frac{3}{2}=0+\frac{3}{2}$
$x^2-\frac{x}{3}=\frac{3}{2}$
The coefficient of the x-term is $-\frac{1}{3}$. Half of $-\frac{1}{3}$ is $-\frac{1}{6}$, and $(-\frac{1}{6})^2$ is $\frac{1}{36}$.
Add $\frac{1}{36}$ to both sides of the equation to complete the square.
$x^2-\frac{x}{3}+\frac{1}{36}=\frac{3}{2}+\frac{1}{36}$
$(x-\frac{1}{6})^2 = \frac{55}{36}$
$x-\frac{1}{6} = ±\sqrt\frac{55}{36}$
$x = \frac{1}{6}±\sqrt\frac{55}{36}$
$x=\frac{1±\sqrt{55}}{6}$
The solutions are $\frac{1±\sqrt{55}}{6}$, and the solution set is {$\frac{1+\sqrt{55}}{6}, \frac{1-\sqrt{55}}{6}$}.
