Solve each problem by constructing and then solving a quadratic equation.
a A rectangular area can be completely tiled with 200 square tiles. If the side length of each tile was
increased by 1 cm, it would take only 128 tiles to tile the area. Find the side length of each tile.

To solve this problem, let's denote the side length of each tile as \( x \) cm.

### Step 1: Construct the Quadratic Equation

1. **Initial Condition:**
   - The total area of the rectangular region is \( 200 \times x^2 \) square cm, where \( x \) is the side length of each tile.
   - Therefore, the area of the rectangular region is \( 200x^2 \) square cm.

2. **After Increasing the Side Length by 1 cm:**
   - The side length of each tile becomes \( x + 1 \) cm.
   - The number of tiles required becomes 128, so the total area of the rectangular region is now \( 128 \times (x + 1)^2 \) square cm.

Since the area of the rectangular region remains the same in both cases, we can equate the two expressions for the area:

\[
200x^2 = 128(x + 1)^2
\]

### Step 2: Solve the Quadratic Equation

1. Expand the right side of the equation:

\[
200x^2 = 128(x^2 + 2x + 1)
\]

2. Distribute 128:

\[
200x^2 = 128x^2 + 256x + 128
\]

3. Move all terms to one side to set the equation to zero:

\[
200x^2 - 128x^2 - 256x - 128 = 0
\]

Simplify the equation:

\[
72x^2 - 256x - 128 = 0
\]

4. Divide the entire equation by 8 to simplify further:

\[
9x^2 - 32x - 16 = 0
\]

### Step 3: Use the Quadratic Formula

The quadratic equation is in the form \( ax^2 + bx + c = 0 \), where \( a = 9 \), \( b = -32 \), and \( c = -16 \).

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Substitute the values of \( a \), \( b \), and \( c \):

\[
x = \frac{-(-32) \pm \sqrt{(-32)^2 - 4(9)(-16)}}{2(9)}
\]

Simplify:

\[
x = \frac{32 \pm \sqrt{1024 + 576}}{18}
\]

\[
x = \frac{32 \pm \sqrt{1600}}{18}
\]

\[
x = \frac{32 \pm 40}{18}
\]

This gives two possible solutions:

\[
x = \frac{72}{18} = 4 \quad \text{or} \quad x = \frac{-8}{18} = -\frac{4}{9}
\]

### Step 4: Interpret the Result

Since the side length \( x \) must be positive, we discard the negative value. Therefore, the side length of each tile is \( x = 4 \) cm.

### Final Answer:
The side length of each tile is **4 cm**.

Would you like any further details or have any questions? Here are some related problems you might find interesting:

1. If the number of tiles increased by 50% instead of 64%, what would the new side length be?
2. How would the area of the entire rectangular region change if the side length increased by 2 cm?
3. Find the perimeter of the rectangular area in the original tiling.
4. If the tiles were triangular instead of square, how would the equation change?
5. What happens to the equation if the increase in the side length is 2 cm instead of 1 cm?

**Tip:** When working with quadratic equations, always check if a simpler form can be obtained by factoring before applying the quadratic formula.

Learn how to solve the problem of tiling a rectangular area with tiles, involving quadratic equations and area calculations. Find the side length of each tile based on given conditions. Suitable for Grades 7-9.

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