Aki's Bicycle Design has determined that when x hundred bicycles are built, the average cost per bicycle =0.9x^2-1.7x+10.861, when C(x) is in hundreds of dollars. How many bicycles should the shop build to minimize the average cost per bicycle?

C(x)=0.9x^2-1.7x+10.861 =
C(x)=0.9(x^2-1.889x)+10.861=

My question is how do they come up with the 1.889 in the equation?

They are using the method of "completing the square" and may be trying to get a perfect square inside the parentheses.

-0.9 x 1.88888... and -1.7 are the same thing. They rounded off the the 1.88888.. to 1.889.

With a complete square inside parentheses, C(x) becomes
0.9(x^2 - 1.889x + 0.8920) + 10.861 - 0.8028
= 0.9(x- 0.9445)^2 + 10.058

You get minimum average cost when x = .9445

To understand how they calculated the 1.889 in the equation, we need to use a concept from algebra called completing the square.

In the equation C(x) = 0.9x^2 - 1.7x + 10.861, we want to rewrite the quadratic function in a form that allows us to find the vertex, which gives us the minimum point of the function. To begin, focus on the quadratic term, 0.9x^2.

To complete the square, we need to take half of the coefficient of the linear term (in this case, -1.7x) and square it. Half of -1.7 is -0.85, and squaring it gives us 0.7225.

So, we rewrite the quadratic term, 0.9x^2, as 0.9(x^2 - 1.889x + 0.7225), where 1.889 is double the value we squared.

Taking this step allows us to factor the quadratic expression inside the parentheses. And to maintain equation equality, we also need to factor the 0.9 out, which gives us:

C(x)=0.9(x^2-1.889x+0.7225)+10.861

Now that we have completed the square, the expression inside the parentheses can be rewritten as a perfect square trinomial.