Question

The side of a square equals teh length of a rectangle. The width of the rectangle is 4 meters longer than its length. The sum of the areas of the square and the rectangle is 48 square centimeters. Find the side of the squarel

Answer 1

Conditions

The side of a square equals the length of a rectangle. The width of the rectangle is 4 meters longer than its length. The sum of the areas of the square and the rectangle is 48 square centimeters. Find the side of the square!

Solution

Let the side of rectangle is $x$. Then the width is $4 + x$.

The area of rectangle will be:

$$x(x + 4)$$

The area of the square is:

$$x^2$$

Then,

$$x(x + 4) + x^2 = 48$$

This is a quadratic equation. Let's solve this using the discriminant formula:

$$2x^2 + 4x - 48 = 0$$

Divide by 2:

$$x^2 + 2x - 24 = 0$$ $$D = b^2 - 4ac = 4 + 4 \cdot 24 = 4 + 96 = 100$$ $$x_{1,2} = \frac{-b \pm \sqrt{D}}{2a} = \frac{-2 \pm 10}{2} = \begin{cases} x_1 = 4 \\ x_2 = -6 \end{cases}$$

$x_2 = -6$ is not a possible value, as the length can't be negative value.

Then, the length of the rectangle is 4. And it's equal to side of the square.

**Answer: 4**