# Question #9c650

Jan 8, 2017

See explanation below:

#### Explanation:

To break even the cost to produce the product, $c \left(x\right)$, must equal the revenue generated by selling the product, $r \left(x\right)$. Therefore,

$c \left(x\right) = r \left(x\right)$

or

$5 x + 18 = 2 {x}^{2}$

We can now solve this for $x$.

$5 x + 18 - \textcolor{red}{5 x} - \textcolor{b l u e}{18} = 2 {x}^{2} - \textcolor{red}{5 x} - \textcolor{b l u e}{18}$

$5 x - \textcolor{red}{5 x} + 18 - \textcolor{b l u e}{18} = 2 {x}^{2} - \textcolor{red}{5 x} - \textcolor{b l u e}{18}$

$0 + 0 = 2 {x}^{2} - 5 x - 18$

$0 = 2 {x}^{2} - 5 x - 18$

We can now factor this.

$2 {x}^{2} - 5 x - 18 = 0$

$\left(2 x - 9\right) \left(x + 2\right) = 0$

We can now solve both of these for $0$:

$2 x - 9 = 0$

$2 x - 9 + 9 = 0 + 9$

$2 x - 0 = 9$

$2 x = 9$

$\frac{2 x}{2} = \frac{9}{2}$

$x = \frac{9}{2}$

and

$x + 2 = 0$

$x + 2 - 2 = 0 - 2$

$x + 0 = - 2$

$x = - 2$

Because you can't produce NEGATIVE units the answer is 9/2 or 4.5 units are needed to be produced to break even.

But because you can't create a 1/2 a unit you need to round up to 5 units.