Question

Solve by using the Quadratic Formula:$\frac{1}{4}{c}^2-\frac{1}{3}c=\frac{1}{12}$

Short answer

The solution of the quadratic equation is$c=\frac{2\pm \sqrt{7}}{3}$

Step-by-step solution

Given information

We are given a quadratic equation$\frac{1}{4}{c}^2-\frac{1}{3}c=\frac{1}{12}$

We multiply by LCD and simplify

We get

$$\begin{gathered} 12(\frac{1}{4}{c}^2-\frac{1}{3}c)=12\frac{1}{12} \\ 3c^2-4c=1 \end{gathered}$$

Subtract 1 from both sides and identify the values of a,b,c

We get,

$3c^2-4c-1=0$

On comparing with standard equation we get$a=3,b=-4,c=-1$

Write quadratic formula and substitute the values

The quadratic formula can be given given as $c=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

On substituting the values we get

$c=\frac{4\pm \sqrt{16-4\left(3\right)(-1)}}{2\left(3\right)}$

Simplify

We get

$$\begin{gathered} c=\frac{4\pm \sqrt{16+12}}{6} \\ c=\frac{4\pm \sqrt{28}}{6} \\ c=\frac{4\pm 2\sqrt{7}}{6} \end{gathered}$$

Take out and remove the common factor

We have,

$$\begin{gathered} c=\frac{4\pm 2\sqrt{7}}{6} \\ c=\frac{2(2\pm \sqrt{7})}{6} \\ c=\frac{2\pm \sqrt{7}}{6} \end{gathered}$$

Conclusion

The solution of the above quadratic equation is$c=\frac{2\pm \sqrt{7}}{3}$