Austin's den is 10 feet longer than it is wide. If the den's area is 144 square feet, what are the dimensions of the room?

Question

This has two answers. Complete the square to solve.

Answer 1

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.

Feel free to reach out to me if you have any further questions!

Answer 2

width (W)

length (L) = 10 + W

Area = L * W = 144

144 = (10 + W) * W

W² + 10W - 144 = 0

Solve for W