Let's set W as the width of the room and L as the length of the den. We know that the den is 10 ft longer than it is wide so that means:
W + 10 = L
Let's call this equation 1.
So also know that the area of the den is 144 ft2. Knowing that the equation for the area of a square is W x L, we know that for this den:
W x L = 144
Let's call this equation 2.
So now we have 2 equations and 2 unknowns. Let's take equation 1 and solve for W:
W = L - 10
Now we can substitute this value into equation 2.
(L - 10) x L = 144
L2 - 10L = 144
This looks very close to a quadratic equation. In fact, if we subtract 144 from both sides, we get the quadratic equation of:
L2 - 10L - 144 = 0
Now we can factor this equation into:
(L+8)(L-18) = 0
That means the two answers for L are -8 ft and 18 ft.
We can now substitute these values for L into equation 1 to solve for W.
W = L - 10
W = 18 - 10 or W = -8 - 10
So our 2 values for W are 8 ft and -18 ft.
Let's just make sure we did this correctly. We know that the area is 144 so (-8*-18) and (18*8) should equal 144 and it does! That means we did it correctly. Give that a den exists, it does not make sense that it has negative values, so the one real answer for this problem is that the width is 8 ft and the length is 18 ft.
