How to solve the following quadratic word problem?

Question

The total cost of carpeting a rectangular room is given the expression $$6x^2 + 18x$$

This is the multiple choice type question so the given options were set up like this.

The length of the room is_______feet, its width is ____ feet and the cost of carpeting is _____ per square. (i'm purposely leaving them as blank because I want to know how to solve it. My question is how would i find the blank parts? if i factor the equation, i'll get $$ 6x(x + 3)$$ this wouldn't give me any information about the blanks above? What would I do?

Answer 1

There are infinite many possible solutions, even assuming that cost per square is constant.

E.g. the length can be $x+3$, width can be $x$ the cost is 6, however it can also be length $1$, width $2x(x+3)$, cost 3.

Answer 2

Steve X is correct that there are infinitely many solutions.

But let's consider what is the "best" solution. It is perfectly natural in a word problem that the unknown $x$ refers to one of the properties of the physical situation (i.e., length or width or price per square foot). Once we factor the polynomial (but see below):

$6 x (x+3)$

we can assume that each of the three terms refers to one of the properties of the physical situation. If $x$ referred to the cost per square foot, it would make no physical sense that the width or length would be $x + 3$. That is, if $x$ is in dollars per square foot, then $x + 3$ would not have the units of a length.

Both length and width have the same units (e.g., feet or meters), so it is most natural that $x$ refers to width and thus $x + 3$ refers to length (which is typically longer than width).

So I think the best solution is:

• Width of room = $x$
• Length of room = $x+3$
• Cost per unit area = 6

Note that there are an infinite number of ways to factor the given equation into three terms, even if we assume that $x$ is a "natural" factor:

$x (x+3) 6$

$x (2 x + 6) 3$

$x (a x + 3 a) (6/a)$ for any real positive $a$.

There are limits on $a$. For example it would make no sense for any property in this problem to have a negative value, and thus $a > 0$. Likewise, it makes no sense for the area of the rug to be extraordinarily large (e.g., larger than the Atlantic Ocean), so there are limits on $a$.

Regardless, as Steve X points out: there are an infinite number of solutions.