An architect has allocated a rectangular space of 589ft^2 for a square dining room and a 12ft wide kitchen. Find both the width of the square dining room and the length of the entire rectangular space.

The width of the square dining room is _____ft
The length of the entire rectangular space is ____ft.

I am going to assume that the kitchen and the dining room have the same width.

so the length of large rectangle is x+12
and the width is x
so x(x+12) = 589
x^2 + 12x = 589
x^2 + 12x + 36 = 589+36
(x+6)^2 = 625
x+6 = ± √625 = ± 25
x = -6 + 25 or -6-25 , the last answer is inadmissible

x = 19

large room is 19 by 31

check:
19 x 31 = 589

Let's assume the width of the square dining room is x ft.

Since the square dining room is square, its length will also be x ft.

According to the given information, the total area of the square dining room and the kitchen is 589 ft^2.

The area of a square is given by the formula: Area = side^2.
So, the area of the dining room will be: x^2 ft^2.

The width of the kitchen is given as 12 ft, and its length will be x ft.

The total area of the dining room and kitchen is the sum of their areas:
x^2 + 12x ft^2.

The sum of these two areas is equal to the total area, which is 589 ft^2:
x^2 + 12x = 589.

To solve this quadratic equation, we can set it equal to zero:
x^2 + 12x - 589 = 0.

Now we can solve this equation using factoring, completing the square, or the quadratic formula.

Using factoring, we find that the factors of -589 that add up to 12 are 41 and -29.

So, the equation can be factored as:
(x + 41)(x - 29) = 0.

Setting each factor equal to zero gives us two solutions:
x + 41 = 0, which gives x = -41 (extraneous solution).
x -29 = 0, which gives x = 29.

Since the width cannot be negative, we can ignore the extraneous solution.

Therefore, the width of the square dining room is 29 ft.

The length of the entire rectangular space is the sum of the length of the square dining room (x) and the width of the kitchen (12 ft).
So, the length of the entire rectangular space is 29 ft + 12 ft = 41 ft.

Therefore, the width of the square dining room is 29 ft, and the length of the entire rectangular space is 41 ft.

To find the width of the square dining room, we need to determine the area of the dining room first. We are given that the area of the entire rectangular space is 589ft^2, so we can subtract the area of the kitchen from it.

The area of the kitchen is given by its width (12ft) multiplied by its length. Since the kitchen is rectangular, we need to solve for its length.

Let's denote the width of the square dining room as x, and the length of the entire rectangular space as y.

The area of the kitchen is: 12ft * y.

To find x, we can start by subtracting the area of the kitchen from the total area of the rectangular space:

589ft^2 - 12ft * y = x^2

Next, we need to find y, the length of the entire rectangular space. We can express y in terms of x and substitute it back into the equation above to solve for x.

Since the square dining room has equal sides, its area is given by x^2. According to the problem, the area of the square dining room plus the area of the kitchen equals the total area:

x^2 + 12ft * y = 589ft^2

Now, we can express y in terms of x by rearranging the equation as:

y = (589ft^2 - x^2) / 12ft

Let's substitute this value of y back into the first equation:

589ft^2 - 12ft * ((589ft^2 - x^2) / 12ft) = x^2

Now we have an equation in terms of x alone. We can simplify it:

589ft^2 - 589ft^2 + x^2 = x^2

This equation simplifies to:

x^2 = 0

This means that x is equal to 0. However, since we are looking for a positive measurement, it implies the dining room has no width, which is impossible.

The given values of 589ft^2 for the total area and 12ft for the kitchen's width do not allow for a possible solution in this case.