Question: 6 Kari uses substitution to decide whether $x^2 + x$ is equivalent to $x(2x + 1)$. She says the expressions are equivalent because they have the same value when $x = 0$. Is Kari correct? Explain.
1. **State the problem:** Kari wants to check if the expressions $x^2 + x$ and $x(2x + 1)$ are equivalent by substituting $x = 0$.
2. **Evaluate both expressions at $x = 0$:**
- For $x^2 + x$: $$0^2 + 0 = 0$$
- For $x(2x + 1)$: $$0(2 \cdot 0 + 1) = 0$$
3. **Check if the expressions are equivalent:**
- Kari sees both expressions equal $0$ at $x=0$ and concludes they are equivalent.
4. **Test equivalence for a different value of $x$ to verify:**
- Let $x = 1$.
- Evaluate $x^2 + x$: $$1^2 + 1 = 1 + 1 = 2$$
- Evaluate $x(2x + 1)$: $$1(2 \cdot 1 + 1) = 1(2 + 1) = 3$$
5. **Since $2 \neq 3$, the expressions are not equivalent.**
6. **Conclusion:** Kari is incorrect because checking equivalence at only one value of $x$ is not sufficient. Expressions must be equal for all values of $x$ to be equivalent.