We are given a rectangle $PQRS$ with side lengths $PS = 20$ cm and $SR = 10 + x + 10 = 20 + x$ cm. A square of side $x$ cm has been cut out from the rectangle. The area of the shaded portion is given as 484 cm$^2$. We need to find the value of $x$.

1. Problem Description

We are given a rectangle $PQRS$ with side lengths $PS = 20$ cm and $SR = 10 + x + 10 = 20 + x$ cm. A square of side $x$ cm has been cut out from the rectangle. The area of the shaded portion is given as 484 cm$^2$. We need to find the value of $x$.

2. Solution Steps

First, we find the area of the rectangle $PQRS$.

The area of the rectangle is given by:

$Area_{rectangle} = length \times width = PS \times SR = 20 \times (20 + x) = 400 + 20x$.

Next, we find the area of the square that was cut out.

The area of the square is given by:

$Area_{square} = x^2$.

The area of the shaded region is the area of the rectangle minus the area of the square.

$Area_{shaded} = Area_{rectangle} - Area_{square}$

$484 = (400 + 20x) - x^2$

$484 = 400 + 20x - x^2$

Rearrange the equation to form a quadratic equation:

$x^2 - 20x + 484 - 400 = 0$

$x^2 - 20x + 84 = 0$

We can solve this quadratic equation using the quadratic formula or by factoring. Let's try factoring:

We need to find two numbers that multiply to 84 and add up to -

2

0. These numbers are -6 and -

1

4. $(x - 6)(x - 14) = 0$

So, the possible values for $x$ are 6 and

1

4.

3. Final Answer

The values of $x$ are 6 and

1

4.

$x = 6$ or $x = 14$

Final Answer: The final answer is $\boxed{6, 14}$