Question

Solve by using the Quadratic Formula:$\frac{1}{9}{d}^2-\frac{1}{2}d=-\frac{1}{3}$

Short answer

The solution of the quadratic equation is$d=\frac{9\pm \sqrt{33}}{4}$

Step-by-step solution

Given information

We are given a quadratic equation$\frac{1}{9}{d}^2-\frac{1}{2}d=-\frac{1}{3}$

Multiply BY LCD and simplify

we get

$$\begin{gathered} 18(\frac{1}{9}{d}^2-\frac{1}{2}d)=18(-\frac{1}{3}) \\ 2d^2-9d=-6 \end{gathered}$$

Add 6 on both sides and identify the values of a,b,c

We get,

$2d^2-9d+6=0$

On comparing with standard equation we get$a=2,b=-9,c=6$

Write quadratic formula and substitute the values in it

The quadratic formula can be given as

$d=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

On substituting the values we get

$d=\frac{9\pm \sqrt{81-4\left(2\right)\left(6\right)}}{2\left(2\right)}$

Simplify

We have,

$$\begin{gathered} d=\frac{9\pm \sqrt{81-48}}{4} \\ d=\frac{9\pm \sqrt{33}}{4} \end{gathered}$$

Conclusion

The solution of the quadratic equation is$d=\frac{9\pm \sqrt{33}}{4}$