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The image depicts a square with side lengths of 10 units. Inside the square, there are two quarter circles that intersect, forming a shape labeled \( x \) whose area we need to find.

### Step-by-Step Solution:

1. **Area of the Square:**
   The square has a side length of 10 units, so its area is:
   \[
   \text{Area of the square} = 10 \times 10 = 100 \text{ square units}
   \]

2. **Area of the Quarter Circles:**
   Each quarter circle has a radius equal to the side of the square, which is 10 units.
   The area of a full circle with radius 10 is:
   \[
   \text{Area of full circle} = \pi \times 10^2 = 100\pi \text{ square units}
   \]
   Since each circle is quartered:
   \[
   \text{Area of one quarter circle} = \frac{100\pi}{4} = 25\pi \text{ square units}
   \]
   Since there are two quarter circles:
   \[
   \text{Total area of both quarter circles} = 2 \times 25\pi = 50\pi \text{ square units}
   \]

3. **Intersection Area (Area of \( x \)):**
   The area of \( x \) is found by subtracting the areas outside the intersection from the total area of the square. The area outside the intersection (shaded areas of quarter circles) can be expressed as:
   \[
   \text{Shaded area} = \text{Area of both quarter circles} - \text{Area of intersection}
   \]

   The area of the intersection is given by:
   \[
   \text{Area of } x = \text{Area of square} - \text{Area of both quarter circles} + \text{Area of intersection}
   \]

4. **Area of the Intersection (Lune Areas):**
   The area of \( x \) involves subtracting the lunes formed by the intersections. The exact formula can be tricky without further geometrical properties or calculus, but in simpler cases, assuming symmetry:
   \[
   \text{Area of } x = \text{Area of the square} - \text{Area of both quarter circles} + \text{Twice the area of the small overlapping region}
   \]
   Given the square and quarter circles, the actual overlapping area (shaded area inside intersection) equals:
   \[
   x = \text{Area of square} - \text{sum of non-overlapping parts}
   \]
   Which after simplification and considering symmetry:
   \[
   x = 100 - 50\pi + \text{intersection area approximated as 2(10)}
   \]

### Final Approximation:
   As the problem simplifies with geometry and estimation,
   \[
   x \approx 100 - 50\pi
   \]
   But adjusting for further calculations (if precisely geometry):
   \[
   x \approx 100 - 50\pi + \text{minor intersection adjustments if needed}
   \]

### Answer:
Given standard techniques for finding the area of x:

\[ x = \text{Area of square} - (\text{Area of both quarter circles} + corrections if any)

Thus, in accurate approximations:

\[ x \approx 100 - 50\pi \text{ units}

Would you like detailed calculations or any further help?  
Here are five questions to extend understanding:

1. What would be the exact method to calculate if the shape was different?
2. How does symmetry help in area calculations?
3. What are real-world applications for calculating areas of complex shapes?
4. How can calculus be applied to calculate overlapping areas accurately?
5. What might be a similar problem but in 3D?

**Tip:** In geometry, visualizing shapes with symmetry often reduces the complexity of the calculations.

Learn how to find the area of x inside a square with intersecting quarter circles using geometric formulas and area subtraction methods. This problem involves calculating the area of a square, the area of quarter circles, and determining the area of intersection using geometric properties.

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