A landscaper has two gardens: one is a square and the other is a rectangle. The width of the rectangular garden is 5 yards less than a side of the square one, and the length of the rectangular garden is 3 yards more than a side of the square garden. If the sum of the areas of both gardens is 165 square yards, find the measure of a side of the square garden.

Mr. Jones wants to plant a garden. The garden is a rectangle 25 yards

wide by 50 yards long. He wants to build a fence along the diagonal of
the rectangle so that he can grow fruit on one side of the diagonal and
vegetables on the other. Calculate the length of the fence. Give your
answer to the nearest yard

Let's denote the side length of the square garden as "x" yards.

The width of the rectangular garden is 5 yards less than a side of the square garden, so its width would be "x - 5" yards.

The length of the rectangular garden is 3 yards more than a side of the square garden, so its length would be "x + 3" yards.

The area of the square garden is given by the formula A = x * x = x^2 square yards.

The area of the rectangular garden is given by the formula A = width * length = (x - 5) * (x + 3) square yards.

The sum of the areas of both gardens is 165 square yards. So, we can set up the equation:

x^2 + (x - 5)(x + 3) = 165

Expanding the equation:

x^2 + (x^2 - 2x - 15) = 165

Combining like terms:

2x^2 - 2x - 150 = 0

Dividing the equation by 2:

x^2 - x - 75 = 0

Now, let's factorize the quadratic equation:

(x - 10)(x + 7.5) = 0

Setting each factor to zero and solving for x:

x - 10 = 0 OR x + 7.5 = 0

x = 10 OR x = -7.5

Since a measurement cannot be negative, the side length of the square garden must be x = 10 yards.

Therefore, the measure of a side of the square garden is 10 yards.

To solve this problem, we need to set up a system of equations based on the information given.

Let's assume that the side length of the square garden is "x" yards.

According to the given information, the width of the rectangular garden is 5 yards less than a side of the square garden. So, the width of the rectangular garden is (x - 5) yards.

Similarly, the length of the rectangular garden is 3 yards more than a side of the square garden. So, the length of the rectangular garden is (x + 3) yards.

Now, we can calculate the areas of both gardens.

The area of the square garden is given by: side length × side length = x × x = x^2 square yards.

The area of the rectangular garden is given by: length × width = (x + 3) × (x - 5) = (x^2 - 2x - 15) square yards.

According to the problem, the sum of the areas of both gardens is 165 square yards, so we can set up the following equation:

x^2 + (x^2 - 2x - 15) = 165

Simplifying the equation, we get:

2x^2 - 2x - 180 = 0

To solve this quadratic equation, we can factorize it as follows:

(2x + 18)(x - 10) = 0

Setting each factor equal to zero and solving for x:

2x + 18 = 0 or x - 10 = 0

2x = -18 or x = 10

Dividing both sides of the first equation by 2, we get:

x = -9 or x = 10

Since the length of a side cannot be negative, the only valid solution for the side length of the square garden is x = 10 yards.

Therefore, the measure of a side of the square garden is 10 yards.

You would let the side length of the square garden be x in yards.

Then the rectangular sides are x-5 and x+3.

Calculate the areas of the two gardens in terms of the side lengths (and x). Equate that to 165 square yards.

You will end up with an equation from which you can solve for x.