Oxponents and Polynomials
Solving a word problem using a quadratic equation with rational roots

The area of a rectangle is $77 \mathrm{~m}^{2}$, and the length of the rectangle is $3 \mathrm{~m}$ more than twice the width. Find the dimensions of the rectangle.
Length : Џm

Step 1 ：The problem is asking for the dimensions of a rectangle given that the area is 77 square meters and the length is 3 meters more than twice the width.

Step 2 ：We know that the area of a rectangle is given by the formula: Area = Length * Width.

Step 3 ：Let's denote the width as x. Then the length would be 2x + 3.

Step 4 ：We can set up the equation as follows: x * (2x + 3) = 77.

Step 5 ：We can solve this quadratic equation to find the value of x, which will give us the width of the rectangle. Once we have the width, we can substitute it into the equation for the length to find the length.

Step 6 ：The solution to the quadratic equation gives two possible values for the width: -7 and 11/2. However, the width of a rectangle cannot be negative, so we discard -7 as a possible solution.

Step 7 ：Therefore, the width of the rectangle is 11/2 meters.

Step 8 ：Substituting this value into the equation for the length, we find that the length of the rectangle is 2*(11/2) + 3 = 11 + 3 = 14 meters.

Step 9 ：Final Answer: The dimensions of the rectangle are \(\boxed{11/2 \text{ m}}\) (width) and \(\boxed{14 \text{ m}}\) (length).