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: \[ \boxed{4x^2 + 14x - 2 = 0} \]