A family has a rectangular back yard that measures 5x+4 by 3x-2 they are building a square patio with side lengths that measure x+3. write an expression for the area of grass that will be left in the back yard after the patio is built. show your work.
The yard would have an area of (5x+4)(3x-2)
The deck would be (x+3)(x+3)
Multiply the binomials and then subtract the one you find for the deck from the one from the yard.
Hint the deck is x^2 + 6x +9
be sure to subtract like terms.
thanks, john
To find the area of grass that will be left in the backyard after the patio is built, you need to calculate the area of the rectangular backyard and subtract the area of the square patio.
The area of the rectangular backyard is given by length x width, so in this case, it would be (5x + 4) * (3x - 2).
The area of the square patio is given by side length squared, so in this case, it would be (x + 3)^2.
To determine the area of grass that will be left in the backyard, subtract the area of the patio from the area of the backyard:
(5x + 4) * (3x - 2) - (x + 3)^2
Now, we can simplify the expression:
Expand the expression for the backyard area:
(5x + 4) * (3x - 2) = 15x^2 + 10x - 8x - 8 = 15x^2 + 2x - 8
Expand the expression for the patio area:
(x + 3)^2 = (x + 3)(x + 3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9
Substitute the expressions for the backyard and patio area into the equation for the area of grass left:
(15x^2 + 2x - 8) - (x^2 + 6x + 9)
Simplify the expression:
15x^2 + 2x - 8 - x^2 - 6x - 9
Combine like terms:
(15x^2 - x^2) + (2x - 6x) + (-8 - 9)
Combine like terms further:
14x^2 - 4x - 17
Therefore, the expression for the area of grass that will be left in the backyard after the patio is built is 14x^2 - 4x - 17.