Answer:
[tex]x^2+14x-51=0[/tex]
Step-by-step explanation:
Since the entire cabinet door is a rectangle, its area is given by:
[tex]A=w\ell[/tex]
The center of the cabinet door measures 12 inches by 16 inches.
And the border of the cabinet door has a width of x inches.
Therefore, the total width of the cabinet door will be (12 + 2x).
Likewise, the total length of the cabinet door will be (16 + 2x).
Thus, the area of the entire cabinet door will be:
[tex]A=(2x+12)(2x+16)[/tex]
We are given that this entire area is 396 square inches. Thus:
[tex]396=(2x+12)(2x+16)[/tex]
Simplify. We can factor out a two from both of the factors on the right-hand side:
[tex]396=2(x+6)(2)(x+8)[/tex]
So:
[tex]396=4(x+6)(x+8)[/tex]
Dividing both sides by four yields:
[tex]99=(x+6)(x+8)[/tex]
Distribute the right-hand side:
[tex]x^2+14x+48=99[/tex]
Therefore, our equation in standard form is:
[tex]x^2+14x-51=0[/tex]
Edit:
We want to find the width of the board. So, we simply have to solve for x. We previously obtained:
[tex]x^2+14x-51=0[/tex]
We can factor:
[tex](x+17)(x-3)=0[/tex]
By the Zero Product Property:
[tex]x+17=0\text{ or } x-3=0[/tex]
Solve for both cases:
[tex]x=-17\text{ or } x=3[/tex]
Length/width cannot be negative. So, we can reject the first solution.
Thus, our only solution is:
[tex]x=3[/tex]
The width of the border of the cabinet door is 3 inches.
This means that the total length of the cabinet is 16 + 2(3) = 22 inches, and the total width of the cabinet is 12 + 2(3) = 18 inches.
