Final: The Rug, Solved

To solve an equation from a word problem, first expand any product. Then move every term to one side so that the other side is zero, factor, and set each factor equal to zero. Finally, check each solution against the story. A width or a length must be positive, so a negative solution is rejected.

For example, a rectangle with width $x$, length $x+2$ and area 15 gives $x\left(x+2\right) = 15$. Expanding gives $x^{2}+2x = 15$, and subtracting 15 from both sides gives $x^{2}+2x-15 = 0$. This factors as $\left(x+5\right)\left(x-3\right) = 0$, so $x = -5$ or $x = 3$. A width cannot be -5, so the width is 3 and the length is 5. Both values solve the equation, but only one of them fits the rectangle.