A rectangular room has an area of 400 sq. ft. The length is 7 ft less than twice the width. Find the dimensions of the room.

w(2w-7) = 400

To solve this problem, we can represent the width of the room as 'x' feet.

According to the given information, the length of the room is 7 feet less than twice the width.

So, the length would be 2x - 7 feet.

The area of a rectangle is calculated by multiplying its length and width.

Given that the area of the room is 400 square feet, we have the equation:

Area = Length × Width
400 sq. ft = (2x - 7) ft × x ft

Now, let's solve the equation step by step:

400 = (2x - 7) × x
400 = 2x² - 7x

To simplify the equation, let's move all the terms to one side and form a quadratic equation:

2x² - 7x - 400 = 0

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

Factoring the quadratic equation, we get:

(2x + 25)(x - 16) = 0

Setting each factor to zero and solving for 'x', we have:

2x + 25 = 0 or x - 16 = 0

For the first case, 2x + 25 = 0, if we solve for 'x', we get:

2x = -25
x = -12.5

Since we're measuring the width of a room, we can't have a negative value.

Therefore, we can conclude that x - 16 = 0:

x = 16

So, the width of the room is 16 feet.

To find the length, we can use the equation we derived earlier:

Length = 2x - 7

Plugging in the value of 'x' (16) into the equation, we find:

Length = 2(16) - 7
Length = 32 - 7
Length = 25

Therefore, the dimensions of the room are:

Width = 16 feet
Length = 25 feet