This problem can be illustrated by a quadratic equation, as follows:

1. Defining the variables:

• Let $x$ be the original side length of the square tile (in cm).

2. Representing the area of the original square:

• The area of a square is given by the formula $\text{Area} = \text{side} \times \text{side}$.
• Therefore, the area of the original tile is $x^2$.
• Given that the area of the original tile is 64 square centimeters, the equation becomes: $x^2 = 64$

3. Representing the area of the new square after increasing the side length by 2 cm:

• If the side length is increased by 2 cm, the new side length becomes $x + 2$ cm.
• The area of the new tile is then $(x + 2)^2$.
• Given that the area of the new tile is 100 square centimeters, the equation becomes: $(x + 2)^2 = 100$

4. Solving the quadratic equation:

• First, solve for the original side length $x$ using the equation $x^2 = 64$: $x = \sqrt{64}$ $x = 8 \text{ cm}$
• Confirming with the second equation: $(x + 2)^2 = 100$ $(8 + 2)^2 = 100$ $10^2 = 100$ $100 = 100 \text{ (True)}$ Thus, the original side length of the tile was 8 cm.

Would you like more details or have any questions about this problem?

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Here are 5 related questions:

1. How would the problem change if the area of the original square was different?
2. What if Tina increased the side length by a different amount, say 3 cm instead of 2 cm?
3. How would you solve the problem if the area of the new square was not given?
4. Can you derive a general formula for the side length if the areas are expressed as variables?
5. How would you interpret the quadratic equation graphically?

Tip: When solving quadratic equations, always check your solution by substituting the value back into the original equation to ensure it satisfies the problem conditions.