The length of a rectangle is 9 centimeters less than four times its width. Its area is 28 square centimeters. Find the dimensions of the rectangle. The width is \( \square \mathrm{cm} \).

Let's denote the width of the rectangle as \( w \) centimeters. Given: 1. The length of the rectangle is 9 centimeters less than four times its width: \( \text{Length} = 4w - 9 \) 2. The area of the rectangle is 28 square centimeters: \( \text{Area} = w \times \text{Length} = 28 \) We can set up the equation for the area of the rectangle: \[ w \times (4w - 9) = 28 \] Now, we can solve this equation to find the width of the rectangle. Solve the quadratic equation by following steps: - step0: Solve by factoring: \(w\left(4w-9\right)=28\) - step1: Expand the expression: \(4w^{2}-9w=28\) - step2: Move the expression to the left side: \(4w^{2}-9w-28=0\) - step3: Factor the expression: \(\left(w-4\right)\left(4w+7\right)=0\) - step4: Separate into possible cases: \(\begin{align}&w-4=0\\&4w+7=0\end{align}\) - step5: Solve the equation: \(\begin{align}&w=4\\&w=-\frac{7}{4}\end{align}\) - step6: Rewrite: \(w_{1}=-\frac{7}{4},w_{2}=4\) The solutions for the width of the rectangle are \( w = -\frac{7}{4} \) or \( w = 4 \) centimeters. Since the width cannot be negative, the width of the rectangle is 4 centimeters. Therefore, the dimensions of the rectangle are: - Width: 4 centimeters - Length: \( 4 \times 4 - 9 = 7 \) centimeters So, the width of the rectangle is 4 centimeters.