Questions: Solving a word problem using a quadratic equation with rational... The length of a rectangle is 5 m less than twice the width, and the area of the rectangle is 52 m^2. Find the dimensions of the rectangle. Length : m Width : m

Solution Steps

To solve this problem, we need to set up a quadratic equation based on the given information. Let the width of the rectangle be \( w \). Then, the length of the rectangle can be expressed as \( 2w - 5 \). The area of the rectangle is given by the product of its length and width, which equals 52 square meters. We can set up the equation \( w(2w - 5) = 52 \) and solve for \( w \).

Let the width of the rectangle be \( w \). The length of the rectangle is given as \( 2w - 5 \). The area of the rectangle is given by the product of its length and width, which equals \( 52 \, \text{m}^2 \). Therefore, we set up the equation: \[ w(2w - 5) = 52 \]

Rearrange the equation to standard quadratic form: \[ 2w^2 - 5w - 52 = 0 \] Solving this quadratic equation, we get the solutions: \[ w = -4 \quad \text{and} \quad w = \frac{13}{2} \] Since the width cannot be negative, we discard \( w = -4 \) and take: \[ w = \frac{13}{2} = 6.5000 \]

Using the width \( w = 6.5000 \), we calculate the length: \[ \text{Length} = 2w - 5 = 2(6.5000) - 5 = 13.0000 - 5 = 8.0000 \]

Final Answer