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Sarah wants to hang a mirror in her room. The mirror and frame must have an area of 15 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, [tex]\( x \)[/tex]?

A. [tex]\( 3x^2 + 14x - 2 = 0 \)[/tex]
B. [tex]\( 4x^2 + 14x - 3 = 0 \)[/tex]
C. [tex]\( x^2 + 12x - 14 = 0 \)[/tex]
D. [tex]\( 2x^2 + 7x - 14 = 0 \)[/tex]

Sagot :

GinnyAnswer
To determine the quadratic equation that represents the thickness of the frame, follow these steps:

1. Identify the dimensions of the mirror:
- Width of the mirror ([tex]\( w \)[/tex]) = 3 feet
- Length of the mirror ([tex]\( l \)[/tex]) = 4 feet

2. Calculate the area of the mirror:
[tex]\[ \text{Area of the mirror} = w \times l = 3 \times 4 = 12 \text{ square feet} \][/tex]

3. Express the total area including the frame:
- The frame adds a uniform thickness ([tex]\( x \)[/tex]) around the mirror.
- Hence, the overall dimensions including the frame would be:
- Width including the frame = [tex]\( w + 2x \)[/tex]
- Length including the frame = [tex]\( l + 2x \)[/tex]

4. Create an equation for the total area (mirror + frame):
[tex]\[ \text{Total area} = (\text{width including the frame}) \times (\text{length including the frame}) \][/tex]
Substituting the dimensions:
[tex]\[ 15 = (3 + 2x) \times (4 + 2x) \][/tex]

5. Expand the equation:
[tex]\[ 15 = (3 + 2x)(4 + 2x) \][/tex]
[tex]\[ 15 = 12 + 6x + 8x + 4x^2 \][/tex]
[tex]\[ 15 = 4x^2 + 14x + 12 \][/tex]

6. Form the quadratic equation:
[tex]\[ 4x^2 + 14x + 12 = 15 \][/tex]
Subtract 15 from both sides to set the equation to zero:
[tex]\[ 4x^2 + 14x + 12 - 15 = 0 \][/tex]
[tex]\[ 4x^2 + 14x - 3 = 0 \][/tex]

Therefore, the quadratic equation that can be used to determine the thickness of the frame [tex]\( x \)[/tex] is:

[tex]\[ \boxed{4x^2 + 14x - 3 = 0} \][/tex]

So, the correct option is:
[tex]\[ 4x^2 + 14x - 3 = 0 \][/tex]
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