Question

The area of a rectangle is 1,056 square inches. Its length is 4 inches longer than 2 times its width.
Which equation can you use to find the width of the rectangle, w?
What is the width of the rectangle?
inches

Answers

GPT-4o mini
Let's define the width of the rectangle as \( w \). According to the problem, the length \( l \) of the rectangle is given by:

\[
l = 2w + 4
\]

The area \( A \) of a rectangle can be calculated using the formula:

\[
A = l \cdot w
\]

Given that the area is 1,056 square inches, we can set up the equation:

\[
w(2w + 4) = 1056
\]

Now, we can expand this equation:

\[
2w^2 + 4w = 1056
\]

To form a standard quadratic equation, we move 1,056 to the left side:

\[
2w^2 + 4w - 1056 = 0
\]

Now we can simplify this equation by dividing everything by 2:

\[
w^2 + 2w - 528 = 0
\]

Now, we can use the quadratic formula to solve for \( w \):

The quadratic formula is given by:

\[
w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In our case, \( a = 1 \), \( b = 2 \), and \( c = -528 \). Plugging in these values:

\[
w = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-528)}}{2 \cdot 1}
\]

\[
w = \frac{-2 \pm \sqrt{4 + 2112}}{2}
\]

\[
w = \frac{-2 \pm \sqrt{2116}}{2}
\]

Calculating the square root:

\[
\sqrt{2116} = 46
\]

Now substituting back into the equation:

\[
w = \frac{-2 \pm 46}{2}
\]

This gives us two potential solutions:

\[
w = \frac{44}{2} = 22 \quad \text{(positive width)}
\]
\[
w = \frac{-48}{2} = -24 \quad \text{(not possible, as width cannot be negative)}
\]

Therefore, the width of the rectangle is:

\[
\boxed{22} \text{ inches}
\]

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