The length of a rectangle is three more than twice its width. The area is 65 m^2. What is its width?

w (2w+3) = 65

2 w^2 + 3 w -65 = 0

w = 20/4 or -26/4
so
w = 5
check
5 (10+3) = 65 yes

To find the width of the rectangle, we'll work through the information given step by step.

Let's assume the width of the rectangle is "w" (in meters).

According to the given information, the length of the rectangle is three more than twice its width. So, the length can be expressed as 2w + 3.

The formula for the area of a rectangle is length multiplied by width. Since we know the area is 65 m², we can write the equation:

(w) * (2w + 3) = 65

Now, we can solve this equation to find the value of "w", which represents the width of the rectangle.

Expanding the equation:

2w² + 3w = 65

Rearranging the equation to form a quadratic equation:

2w² + 3w - 65 = 0

To solve this quadratic equation, we can either factorize it or use the quadratic formula. In this case, let's use the quadratic formula:

The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x can be found using the formula:

x = (-b ± √(b² - 4ac)) / 2a

For our equation 2w² + 3w - 65 = 0, the values of a, b, and c are:
a = 2
b = 3
c = -65

Substituting these values into the quadratic formula:

w = (-3 ± √(3² - 4 * 2 * -65)) / (2 * 2)

Simplifying:

w = (-3 ± √(9 + 520)) / 4
w = (-3 ± √529) / 4

Now, we have two possible solutions for the value of "w":

w₁ = (-3 + √529) / 4
w₂ = (-3 - √529) / 4

Evaluating the solutions:

w₁ = (24) / 4 = 6
w₂ = (-29) / 4 = -7.25

Since the width of a rectangle cannot be negative, we discard the negative solution. Therefore, the width of the rectangle is 6 meters.