Question

The cost to produce a product is modeled by the function f(x) = 5x2 - 70x + 258 where x is the number of products produced. Complete the square to determine the minimum cost of producing this product. (2 points)
Select one:
a. 5(x - 7)2 + 13; The minimum cost to produce the product is $13.
b. 5(x - 7)2 + 13; The minimum cost to produce the product is $7.
c. 5(x - 7)2 + 258; The minimum cost to produce the product is $7.
d. 5(x - 7)2 + 258; The minimum cost to produce the product is $258.

Answer 1

Can you help me here @SolomonZelman

Answer 2

\(\large\color{black}{ \displaystyle f(x) = 5x^2 - 70x + 258 }\)
\(\large\color{black}{ \displaystyle f(x) = 5\left(x^2 - 14x\right) + 258 }\)

TELL ME:
what number would you want to add to \(x^2-14x\),
to make \(x^2-14x\) a perfect square trinomial?

Answer 3

258? i don't know

Answer 4

I will attempt to familiarize you with the following rule:

\(\large\color{black}{ \displaystyle \left({\rm \color{blue}{b}}+{\rm \color{red}{a}}\right)^2=\left({\rm \color{blue}{b}}+{\rm \color{red}{a}}\right)\left({\rm \color{blue}{b}}+{\rm \color{red}{a}}\right)= {\rm \color{blue}{b}}^2+2{\rm \color{red}{a}}{\rm \color{blue}{b}}+{\rm \color{red}{a}}^2}\)

Answer 5

Same way,

\(\large\color{black}{ \displaystyle \left({\rm \color{blue}{b}}-{\rm \color{red}{a}}\right)^2=\left({\rm \color{blue}{b}}-{\rm \color{red}{a}}\right)\left({\rm \color{blue}{b}}-{\rm \color{red}{a}}\right)= {\rm \color{blue}{b}}^2-2{\rm \color{red}{a}}{\rm \color{blue}{b}}+{\rm \color{red}{a}}^2}\)

Answer 6

that rule makes sence

Answer 7

So you have the
\(\large\color{black}{ {\rm \color{blue}{b}}^2-2{\rm \color{red}{a}}{\rm \color{blue}{b}}}\)
so far,.

and that in your case is:
\(\large\color{black}{ {\rm \color{blue}{x}}^2-2{\rm \color{red}{\cdot 7}}{\rm\cdot \color{blue}{x}}}\)

Answer 8

remains is the \(\large\color{black}{ {\rm \color{red}{a}}^2}\), and in your case, that is: \(\large\color{black}{ {\rm \color{red}{7}}^2=49}\)

Answer 9

oh sorry, i typed it wrong, it is f(x) = 5x^2 - 70x + 258

that 2 is an exponent, my bad

Answer 10

\(\large\color{black}{ \displaystyle f(x) = 5\left(x^2 - 14x\right) + 258 }\)

\(\large\color{black}{ \displaystyle f(x) = 5\left(x^2 - 14x+49-49\right) + 258 }\)

\(\large\color{black}{ \displaystyle f(x) = 5\left(x^2 - 14x+49\right)-49\times 5 + 258 }\)

Answer 11

and yes, I assumed that from the ver beginning so it's all good

Answer 12

ok good :)

Answer 13

\(\large\color{black}{ \displaystyle f(x) = 5\left(x^2 - 14x+49\right)-245 + 258 }\)

\(\large\color{black}{ \displaystyle f(x) = 5\left(x^2 - 14x+49\right) + 13 }\)

\(\large\color{black}{ \displaystyle f(x) = 5\left(x-7\right)^2 + 13 }\)

Answer 14

the minimum is at x=7, and it equivalent to?

Answer 15

where did you get 14?

Answer 16

I just did the completing the square method, that is all.
If you don't know what I did, then I guess some online resources would be helpful.

Answer 17

ok so it is equivalent to the minimum cost to produce, right?

Answer 18

oh wait, i see where you gut the -14 from!

Answer 19

ok, now this all makes sense! thank you so much for your help @SolomonZelman !!!