# Problem of the Week

## Updated at Sep 7, 2020 12:31 PM

For this week we've brought you this equation problem.

How would you solve the equation $$\frac{3(3-z)}{{(\frac{5}{z})}^{2}}=-6$$?

Here are the steps:

$\frac{3(3-z)}{{(\frac{5}{z})}^{2}}=-6$

1
Use Division Distributive Property: $${(\frac{x}{y})}^{a}=\frac{{x}^{a}}{{y}^{a}}$$.
$\frac{3(3-z)}{\frac{{5}^{2}}{{z}^{2}}}=-6$

2
Simplify  $${5}^{2}$$  to  $$25$$.
$\frac{3(3-z)}{\frac{25}{{z}^{2}}}=-6$

3
Invert and multiply.
$3(3-z)\times \frac{{z}^{2}}{25}=-6$

4
Simplify  $$3(3-z)\times \frac{{z}^{2}}{25}$$  to  $$\frac{3(3-z){z}^{2}}{25}$$.
$\frac{3(3-z){z}^{2}}{25}=-6$

5
Regroup terms.
$\frac{3{z}^{2}(3-z)}{25}=-6$

6
Multiply both sides by $$25$$.
$3{z}^{2}(3-z)=-150$

7
Expand.
$9{z}^{2}-3{z}^{3}=-150$

8
Move all terms to one side.
$9{z}^{2}-3{z}^{3}+150=0$

9
Factor out the common term $$3$$.
$3(3{z}^{2}-{z}^{3}+50)=0$

10
Factor $$3{z}^{2}-{z}^{3}+50$$ using Polynomial Division.
$3(-{z}^{2}-2z-10)(z-5)=0$

11
Solve for $$z$$.
$z=5$

12
$z=\frac{2+6\imath }{-2},\frac{2-6\imath }{-2}$
$z=5,\frac{2+6\imath }{-2},\frac{2-6\imath }{-2}$
$z=5,-1-3\imath ,-1+3\imath$ 