Give situations illustrates quadratic equations by representing each situation with a mathematical sentence



Tina is designing a square shaped tile with a border. the area of the tile is 64 square centimeters. if she increases the side length of the square by 2cm, ghe new area is 100 square centimeters. what was the original side length of the tile?

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.

Learn how to solve Tina's square tile problem involving quadratic equations. Understand how to calculate areas, solve quadratic equations, and interpret geometric changes. Suitable for Grades 7-9 students learning algebra and geometry.

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