Question

Solve the equation that models the volume of the shipping box, 8(n + 2)(n + 4) = 1,144. If you get two solutions, are they both reasonable?

Answer 1

`n=-15\text{ or }n=9`

Only n=9 is reasonable.

Explanation:

We have been given that the volume of the shipping box is:
`8(n + 2)(n + 4) = 1,144`.

First of all we will distribute 8 to (n+2).

`(8n+16)(n+4) = 1,144`

Using FOIL we will get,

`8n^2+32n+16n+64= 1,144`

`8n^2+48n+64= 1,144`

`8n^2+48n+64-1,144=0`

`8n^2+48n-1080=0`

Upon dividing our equation by 8 we will get,

`n^2+6n-135=0`

Now we will factor out our given equation by splitting the middle term.

`n^2+15n-9n-135=0`

`n(n+15)-9(n+15)=0`

`(n+15)(n-9)=0`

`(n+15)=0\text{ or }(n-9)=0`

`n=-15\text{ or }n=9`

Since the volume of a box can not be negative, therefore, only one solution is reasonable, that is n=9.

Answer 2

`n=9\\n=-15`

`n=9` is reasonable.

`n=-15` is not reasonable, because the volume of the shipping box can't be negative.

Explanation:

1. Apply the Distributive property to the given equation. Then, you have:

`8n^(2)+32n+16n+64=1,144\\8n^(2)+48n-1080=0`

2. You obtain a quadratic equation, then you can use the quadratic formula to solve it:

`n=\frac{-b+/-\sqrt{b^(2)-4ac}}{2a}\\a=8\\b=48\\c=-1080`

3. Then, you obtain:

`n=\frac{-48+/-\sqrt{(48)^(2)-4(8)(-1080)}}{2(8)}\\n=9\\n=-15`

4. The volume of the shipping box can't be negative. Therefore,
`n=9` is reasonable and
`n=-15` is not reasonable.