Your furniture store sells two types of dining room tables. The first, type A, costs $208 and you make a $24 profit on each one. The second, type B, costs $105 and you make a $15 profit on each one. You can order no more than 190 tables this month, and you need to make at least $3750 profit on them. If you must order at least one of each type of table, how many of each type of table should you order if you want to minimize your cost?

x=\tealD kx=kx, equals, start color #01a995, k, end color #01a995 is a root, or solution, of the equation f(x)=0f(x)=0f, left parenthesis, x, right parenthesis, equals, 0

\tealD kkstart color #01a995, k, end color #01a995 is a zero of function fff

(\tealD k,0)(k,0)left parenthesis, start color #01a995, k, end color #01a995, comma, 0, right parenthesis is an xxx-intercept of the graph of y=f(x)y=f(x)y, equals, f, left parenthesis, x, right parenthesis

x-\tealD kx−kx, minus, start color #01a995, k, end color #01a995 is a linear factor of f(x)f(x)f, left parenthesis, x, right parenthesis

Let's understand this with the polynomial g(x)=(x-3)(x+2)g(x)=(x−3)(x+2)g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, which can be written as g(x)=(x-3)(x-(-2))g(x)=(x−3)(x−(−2))g, left parenthesis, x, right parenthesis, equals, left parenthesis, x, minus, 3, right parenthesis, left parenthesis, x, minus, left parenthesis, minus, 2, right parenthesis, right parenthesis.

First, we see that the linear factors of g(x)g(x)g, left parenthesis, x, right parenthesis are (x-\tealD3)(x−3)left parenthesis, x, minus, start color #01a995, 3, end color #01a995, right parenthesis and (x-(\tealD{-2}))(x−(−2))left parenthesis, x, minus, left parenthesis, start color #01a995, minus, 2, end color #01a995, right parenthesis, right parenthesis.

If we set g(x)=0g(x)=0g, left parenthesis, x, right parenthesis, equals, 0 and solve for xxx, we get x=\tealD3x=3x, equals, start color #01a995, 3, end color #01a995 or x=\tealD{-2}x=−2x, equals, start color #01a995, minus, 2, end color #01a995. These are the solutions, or roots, of the equation.

A zero of a function is an xxx-value that makes the function value 000. Since we know x=3x=3x, equals, 3 and x={-2}x=−2x, equals, minus, 2 are solutions to g(x)=0g(x)=0g, left parenthesis, x, right parenthesis, equals, 0, then \tealD33start color #01a995, 3, end color #01a995 and \tealD{-2}−2start color #01a995, minus, 2, end color #01a995 are zeros of the function ggg.

Finally, the xxx-intercepts of the graph of y=g(x)y=g(x)y, equals, g, left parenthesis, x, right parenthesis satisfy the equation 0=g(x)0=g(x)0, equals, g, left parenthesis, x, right parenthesis, which was solved above. The xxx-intercepts of the equation are

Step-by-step explanation: