A family has a rectangular backyard that measures 2x + 3x by 3x - 5. they are building a square patio with side lengths that measure x + 2. write an expression for the area of grass that will be left in the backyard after the patio is built
that would be the area of the yard less the area of the patio. So,
(2x+3x)(3x-5)-(x+2)^2
or, assuming a typo,
(2x+3)(3x-5)-(x+2)^2
= (6x^2-x-15)-(x^2+4x+4)
= 5x^2-5x-19
To find the area of grass that will be left in the backyard after the patio is built, we need to calculate the area of the backyard and subtract the area of the patio.
The given dimensions of the backyard are 2x + 3x by 3x - 5, which means the length is 2x + 3x and the width is 3x - 5.
Area of the backyard = length * width = (2x + 3x) * (3x - 5) = 5x * (3x - 5) = 15x^2 - 25x
The dimensions of the square patio are given as x + 2, thus the area of the patio would be:
Area of the patio = side length * side length = (x + 2) * (x + 2) = (x + 2)^2 = x^2 + 4x + 4
To find the area of grass that will be left in the backyard, we subtract the area of the patio from the area of the backyard:
Area of grass = Area of backyard - Area of patio
Area of grass = (15x^2 - 25x) - (x^2 + 4x + 4)
Area of grass = 15x^2 - 25x - x^2 - 4x - 4
Area of grass = 14x^2 - 29x - 4
Therefore, the expression for the area of grass that will be left in the backyard after the patio is built is 14x^2 - 29x - 4.