We are told that the length of a rectangle is 12 feet more than twice the width.
If we call L the lengthand W the width of the rectangle, we cn write the following equation expressing the info given:
L = 2 W + 12
we are also told that the area of the rectangle (which by the way is expressed as length times width) is 320 square feet.
The we can write a second equation as:
L * W = 320
Now, we can replace the length "L" with the expression we have in the first equation above, and re-write the equation for the area as:
L * W = 320
(2 W + 12) * W = 320
Use distributive property to eliminate parenthesis:
2 W^2 + 12 W = 320
2 W^2 + 12 W - 320 = 0
We now extract "2" as common factor:
2 (W^2 + 6 W - 160) = 0
and try to factor out the quadratic expression inside the parenthesis:
w^2 + 6 W - 160 = W^2 + 16 W - 10 W - 160 =
Factoring by grouping:
W^2 + 16 W - 10 W - 160 = W (W + 16) - 10 (W + 16) = (W + 16) (W - 10)
Now we have the factored out quadratic equal to zero:
2 (W + 16) (W - 10) = 0
Since this is a product equal zero, that means that the factors must be zero to produce a zero as a result.
That is (W + 16) = 0 which leads to W = -16
or W - 10 = 0 which leads to W = 10
since we are looking for a positive number (a width) then we pick the positive answer W = 10
Now, knowing that W is 10 feet long, we can find the value of the rectangle's length using the first equation :
L = 2 (10) + 12 = 20 + 12 = 32
Then L = 32 feet and W = 10 feet