An architect has allocated a rectanglular space of 646ft^2 for a square dining room and a 15ft wide kitchen. Find both the width of the square dining room and the length of the entire rectangular space.

looks like

x^2 + 15x = 646
(x-19)(x+34) = 0
So, the dining room is 19x19 and the kitchen is 19x15

rectangular space is 19x34

To find the width of the square dining room, we can start by assuming that the width and length of the dining room are the same, as it is a square. Let's represent this unknown width with the variable "x".

The area of a square is given by the formula: area = side^2, which means the area of the dining room is x^2.

Now, we know that the architect has allocated a rectangular space with an area of 646ft^2 for both the dining room and the kitchen together. The kitchen has a width of 15ft, which means the length of the dining room plus the width of the kitchen equals the length of the entire rectangular space.

We can represent the length of the entire rectangular space with the variable "y".

Therefore, we can set up the equation: x^2 + 15x = 646.

To solve this quadratic equation, we can rearrange it to:

x^2 + 15x - 646 = 0.

Now, we can factor the quadratic equation or use the quadratic formula to find the values of "x". By factoring or using the quadratic formula, we get two possible values for "x": x = 14 and x = -46.

Since the width of a room cannot be negative, we discard the value of x = -46.

Therefore, the width of the square dining room is x = 14ft.

To find the length of the entire rectangular space, we can substitute the value of x into the equation for the length: y = x + 15.

y = 14 + 15

y = 29.

Therefore, the length of the entire rectangular space is 29ft.

To summarize:

Width of the square dining room = 14ft
Length of the entire rectangular space = 29ft.