Let me analyze the problem provided in the image and solve it for you.

Problem Overview:

The image describes a geometric problem involving two overlapping squares. The shaded region's area is given as $51 \, \text{cm}^2$. The task is to find the areas of both the smaller square and the larger square, given that the smaller square's side length is $3 \, \text{cm}$ less than the larger square.

Solution:

Let the side length of the larger square be $x \, \text{cm}$. Then:

1. Smaller square's side length = $x - 3 \, \text{cm}$.
2. Area of the larger square = $x^2 \, \text{cm}^2$.
3. Area of the smaller square = $(x - 3)^2 \, \text{cm}^2$.
4. Shaded area = Area of the larger square - Area of the smaller square.

This leads to the equation: $x^2 - (x - 3)^2 = 51$

Expand $(x - 3)^2$:

$(x - 3)^2 = x^2 - 6x + 9$

Substitute this back into the equation: $x^2 - \big(x^2 - 6x + 9\big) = 51$

Simplify: $x^2 - x^2 + 6x - 9 = 51$ $6x - 9 = 51$

Solve for $x$: $6x = 60$ $x = 10$

Results:

1. Side length of the larger square: $x = 10 \, \text{cm}$.
2. Side length of the smaller square: $x - 3 = 7 \, \text{cm}$.
3. Area of the larger square: $10^2 = 100 \, \text{cm}^2$.
4. Area of the smaller square: $7^2 = 49 \, \text{cm}^2$.

Final Verification:

Shaded area = $100 - 49 = 51 \, \text{cm}^2$, which matches the given shaded area.

Answer:

• Area of the larger square: $100 \, \text{cm}^2$
• Area of the smaller square: $49 \, \text{cm}^2$

Would you like a detailed explanation of the steps or related concepts?

Expanding on this solution:

Here are 5 questions to further your understanding:

1. How does the formula for the area of a square relate to its side length?
2. What is the importance of expanding $(x - 3)^2$ in solving this problem?
3. Could this problem be solved graphically? How?
4. What happens to the shaded area if the difference between the squares' side lengths increases?
5. How could you generalize this problem to include rectangles instead of squares?

Tip: Always double-check your final answers by plugging them back into the original problem for verification!