A carpenter is putting a skylight in a roof. If the roof measures 4x + 5 by 7x + 7 and the skylight measures 10x + 10 by 2x + 6, what is the area of the remaining roof after the skylight is built. Put your answer in factored form.
To find the area of the remaining roof after the skylight is built, we need to subtract the area of the skylight from the total area of the roof.
Let's find the total area of the roof first.
The length of the roof is given by 4x + 5, and the width of the roof is given by 7x + 7.
So, the area of the roof is (4x + 5) * (7x + 7).
Now let's find the area of the skylight.
The length of the skylight is given by 10x + 10, and the width of the skylight is given by 2x + 6.
So, the area of the skylight is (10x + 10) * (2x + 6).
To find the area of the remaining roof, we subtract the area of the skylight from the total area of the roof:
Remaining Area = Total Area of the Roof - Area of the Skylight
Remaining Area = (4x + 5) * (7x + 7) - (10x + 10) * (2x + 6)
To simplify this expression, we can use the distributive property and then combine like terms. Let's go through the steps:
1. Distribute:
Remaining Area = (28x^2 + 28x + 35x + 35) - (20x^2 + 60x + 20x + 60)
2. Combine like terms:
Remaining Area = 28x^2 + 63x + 35 - 20x^2 - 80x - 60
3. Combine like terms again:
Remaining Area = (28x^2 - 20x^2) + (63x - 80x) + (35 - 60)
4. Simplify:
Remaining Area = 8x^2 - 17x - 25
So, the area of the remaining roof after the skylight is built is 8x^2 - 17x - 25, in factored form.
To find the area of the remaining roof after the skylight is built, we need to subtract the area of the skylight from the area of the roof.
The area of the roof is given by its length multiplied by its width: (4x + 5) * (7x + 7).
The area of the skylight is given by its length multiplied by its width: (10x + 10) * (2x + 6).
So, the area of the remaining roof is: (4x + 5) * (7x + 7) - (10x + 10) * (2x + 6).
Let's simplify this expression:
(4x + 5) * (7x + 7) - (10x + 10) * (2x + 6)
= 28x^2 + 28x + 35x + 35 - 20x^2 - 60x + 70x + 210
= 28x^2 + 63x + 35 - 20x^2 + 10x^2 + 130x + 210
= (28x^2 - 20x^2 + 10x^2) + (63x + 130x) + (35 + 210)
= 18x^2 + 193x + 245.
So, the factored form of the area of the remaining roof is: 18x^2 + 193x + 245.
(4x + 5)(7x + 7) = 28x^2+63x+35
(10x+10)(2x + 6) = 20x^2+80x+60
---------------subtract----------
8x^2 -17 x -25 = remaining roof
which is
(8x-25)(x+1)