Math Problem Statement

Which expression represents the area of the shaded region?

Solution

The problem asks for an expression representing the area of the shaded region. Here's how we can solve this:

  1. Area of the large square: The large square has a side length of xx. The area AsquareA_{\text{square}} of this square is: Asquare=x2A_{\text{square}} = x^2

  2. Area of the smaller rectangle: The small rectangle has dimensions 5 (height) and 2 (width). The area ArectangleA_{\text{rectangle}} of the rectangle is: Arectangle=5×2=10A_{\text{rectangle}} = 5 \times 2 = 10

  3. Area of the shaded region: The shaded region is the area of the large square minus the area of the small rectangle: Ashaded=AsquareArectangle=x210A_{\text{shaded}} = A_{\text{square}} - A_{\text{rectangle}} = x^2 - 10

Thus, the expression representing the area of the shaded region is: x210\boxed{x^2 - 10}

Would you like more details or have any questions? Here are some related questions to expand the topic:

  1. How would the expression change if the dimensions of the small rectangle were different?
  2. What is the area of the shaded region if x=6x = 6?
  3. How would you express the perimeter of the large square in terms of xx?
  4. What would happen to the shaded area if the rectangle's width doubled?
  5. How can this problem be generalized to any square with a smaller rectangle subtracted from it?

Tip: Always subtract the areas of removed parts when calculating the area of complex shapes.

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Math Problem Analysis

Mathematical Concepts

Geometry
Algebra
Area Calculation

Formulas

Area of a square = side^2
Area of a rectangle = length × width

Theorems

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Suitable Grade Level

Grades 7-9