The problem describes a rectangle where the length is $x$ cm. The width is $x/2 + 3$ cm. The area of the rectangle is $360$ cm$^2$. We need to form a quadratic equation in terms of $x$ and solve it to find the length and width of the rectangle.

1. Problem Description

The problem describes a rectangle where the length is $x$ cm. The width is $x/2 + 3$ cm. The area of the rectangle is $360$ cm$^2$. We need to form a quadratic equation in terms of $x$ and solve it to find the length and width of the rectangle.

2. Solution Steps

Let the length of the rectangle be $x$ cm.

The width of the rectangle is given as $x/2 + 3$ cm.

The area of a rectangle is given by the formula:

$Area = length \times width$

In this case, the area is $360$ cm$^2$, so we have:

$360 = x \times (x/2 + 3)$

Multiplying both sides by 2 to get rid of the fraction:

$720 = x(x+6)$

Expanding the right side:

$720 = x^2 + 6x$

Rearranging into a quadratic equation:

$x^2 + 6x - 720 = 0$

Now we need to solve this quadratic equation. We can use factoring:

We are looking for two numbers that multiply to $-720$ and add up to $6$. Those numbers are $30$ and $-24$.

So, the equation can be factored as:

$(x + 30)(x - 24) = 0$

This gives us two possible solutions for $x$:

$x + 30 = 0$ or $x - 24 = 0$

$x = -30$ or $x = 24$

Since length cannot be negative, we take $x = 24$.

So, the length of the rectangle is $24$ cm.

The width is $x/2 + 3 = 24/2 + 3 = 12 + 3 = 15$ cm.

3. Final Answer

The length of the rectangle is 24 cm and the width of the rectangle is 15 cm.