How do you interpret the negative solution?

Question

A rectangle is such that its area is $308 \,m^2$.

Its length is $8\, m$ longer than its width.

What is its length?

Let $x$ denote its width. Its length is then $x+8$ and $x$ satisfies

\begin{align} x(x+8)&=308\\ x^2+8x-308&=0\\ (x+22)(x-14)&=0\\ x=-22 \quad \text{or} \quad x&=14. \end{align}

We select the positive value of $x$, the width is $14\, m$, we have $14+8=22$, the length is $22\,m$.

Question. How do you interpret $x=-22$?

Answer 1

Consider the following problem: Find the rectangle with corners $(0,0),(x,0),(x,y),(y,0)$ such that its area is $308$ and $x=y+8$.

This leads to the equation

$$308=|x||y|=|xy|=|y(y+8)|.$$ Resolving the absolute value we get two equations $$y^2+8y-308=0,\quad y^2+8y+308 = 0.$$ The second one has no real root, so you are left with exactly the equation you encountered in your solution only that in this case the negative values for $x,y$ also make sense.

Answer 2

If we reformulate the question that you are asking in terms of integrals (since integrals and area are related) we could say that we are looking for the solution of

$$\int^{x+8}_0 x\,dt = 308 $$ There are indeed two solution, one which is $x=-22$. We have $x$ is negative and $x+8$ is negative, which means that it might be convenient to write $$\int^{x+8}_0 x\,dt = \int^0_{x+8} - x \,dt$$ which we can interpret easily. We have that it is the rectangle on the left side of the vertical axis which also has the same area.