Question

Tom wants to hang a mirror in his room. The mirror and frame must have an area of 14 square feet. The mirror is 3 feet wide and 4 feet long. Which quadratic equation can be used to determine the thickness of the frame, x?

Square with an inner frame with height of 3 ft on the left frame and width of 4 ft on the top. Arrow on the bottom frame with an x and an arrow on the right frame with an x.

4x2 + 14x − 2 = 0
3x2 + 10x − 14 = 0
2x2 + 12x − 14 = 0
x2 + 7x − 14 = 0

Answer 1

To determine the thickness of the frame \( x \), we start by analyzing the problem.

The mirror has fixed dimensions:
- Width = 3 feet
- Length = 4 feet

Thus, the area of the mirror itself is:
\[
\text{Area} = \text{Width} \times \text{Length} = 3 \times 4 = 12 \text{ square feet}
\]

Let \( x \) be the thickness of the frame. When we add the frame around the mirror, the overall dimensions of the mirror plus frame become:
- Width = \( 3 + 2x \) (adding \( x \) on both sides)
- Length = \( 4 + 2x \) (adding \( x \) on both sides)

The area of the frame plus the mirror is then given by:
\[
\text{Total Area} = (3 + 2x)(4 + 2x)
\]

We know that the total area must equal 14 square feet:
\[
(3 + 2x)(4 + 2x) = 14
\]

Now we can expand the left side:
\[
12 + 6x + 8x + 4x^2 = 14
\]
\[
12 + 14x + 4x^2 = 14
\]

Subtracting 14 from both sides gives:
\[
4x^2 + 14x + 12 - 14 = 0
\]
\[
4x^2 + 14x - 2 = 0
\]

Thus, the quadratic equation that can be used to determine the thickness of the frame \( x \) is:
\[
\boxed{4x^2 + 14x - 2 = 0}
\]