A third of the square of a number, increased by 2

To express "a third of the square of a number, increased by 2," you can follow these steps:

1. **Square the number**: Let the number be \( x \). The square of the number is \( x^2 \).
2. **Find a third of the square**: A third of the square is \( \frac{1}{3} \times x^2 = \frac{x^2}{3} \).
3. **Increase it by 2**: Finally, add 2 to the result, giving \( \frac{x^2}{3} + 2 \).

So, the expression is:

\[
\frac{x^2}{3} + 2
\]

Would you like more details or have any questions?

Here are some related questions:

1. What is the value of the expression when \( x = 3 \)?
2. How do you simplify an expression like \( \frac{x^2}{3} + 2 \) for different values of \( x \)?
3. What is the derivative of \( \frac{x^2}{3} + 2 \)?
4. How do you graph the function \( f(x) = \frac{x^2}{3} + 2 \)?
5. What happens if you subtract 2 instead of adding 2?
6. Can you calculate \( \frac{x^2}{3} + 2 \) for a negative value of \( x \)?
7. What is the vertex of the parabola defined by \( f(x) = \frac{x^2}{3} + 2 \)?
8. How do you find the roots of the equation \( \frac{x^2}{3} + 2 = 0 \)?

**Tip:** When working with expressions involving squares, remember that squaring always results in a non-negative value, regardless of whether the original number is positive or negative.

Explore how to calculate 'a third of the square of a number, increased by 2' with step-by-step guidance. This problem involves arithmetic operations, algebraic manipulation, and division by a fraction. Suitable for students in Grades 7-9.

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