Question

The cost to produce a product is modeled by the function f(x) = 5x^2 − 70x + 258, where x is the number of products produced. Complete the square to determine the minimum cost of producing this product.
5(x − 7)^2 + 13; The minimum cost to produce the product is $13.
5(x − 7)^2 + 13; The minimum cost to produce the product is $7.
5(x − 7)^2 + 258; The minimum cost to produce the product is $7.
5(x − 7)^2 + 258; The minimum cost to produce the product is $258.

Answer 1

The minimum of a quadratic function, with a positive coefficient a, is its vertex.

Let's find the x₀ coordinate.

$f(x) = 5x^2 -70x + 258\\\\x_0=\dfrac{-b}{2a}=\frac{-(-70)}{2*5} =\frac{70}{10} =7$

Now we need to find y₀ coordinate. That will be the minimum of function.

$y_0=5\times7^2-70\times7+258=13$

So, the minimum cost to produce the product is $13

Decompose 5x^2 − 70x + 258 into multipliers

$5x^2 - 70x + 258=(5x^2-70x+245)+13=5(x^2-14+49)+13=\\=5(x-7)^2+13$

Answer: 5(x − 7)^2 + 13; The minimum cost to produce the product is $13.