Kari uses substitution to decide whether [tex]x^2 + x[/tex] is equivalent to [tex]x(2x + 1)[/tex]. She says the expressions are equivalent because they have the same value when [tex]x = 0[/tex].

Is Kari correct? Explain your answer.



Answer :

Kari is incorrect in her conclusion that [tex]\(x^2 + x\)[/tex] is equivalent to [tex]\(x(2x + 1)\)[/tex] simply because the two expressions have the same value when [tex]\(x = 0\)[/tex]. To determine if two expressions are equivalent, they must yield the same value for all values of [tex]\(x\)[/tex], not just a single value.

Let's examine the given expressions step-by-step for [tex]\(x = 0\)[/tex] and other values.

1. Evaluate [tex]\(x^2 + x\)[/tex] when [tex]\(x = 0\)[/tex]:

[tex]\[ x^2 + x = 0^2 + 0 = 0 \][/tex]

2. Evaluate [tex]\(x(2x + 1)\)[/tex] when [tex]\(x = 0\)[/tex]:

[tex]\[ x(2x + 1) = 0(2 \cdot 0 + 1) = 0 \cdot 1 = 0 \][/tex]

Indeed, both expressions have the same value of 0 when [tex]\(x = 0\)[/tex].

3. Check if the expressions are equivalent for other values of [tex]\(x\)[/tex]:

Let's choose another value, say [tex]\(x = 1\)[/tex]:

- For [tex]\(x^2 + x\)[/tex]:

[tex]\[ x^2 + x = 1^2 + 1 = 1 + 1 = 2 \][/tex]

- For [tex]\(x(2x + 1)\)[/tex]:

[tex]\[ x(2x + 1) = 1(2 \cdot 1 + 1) = 1(2 + 1) = 1 \cdot 3 = 3 \][/tex]

When [tex]\(x = 1\)[/tex], the expression [tex]\(x^2 + x\)[/tex] evaluates to 2, while [tex]\(x(2x + 1)\)[/tex] evaluates to 3.

4. Compare the structures of the expressions:

We can further verify by expanding and simplifying both expressions generally:

- [tex]\(x^2 + x\)[/tex] is already in its simplest form.
- Expanding [tex]\(x(2x + 1)\)[/tex]:

[tex]\[ x(2x + 1) = 2x^2 + x \][/tex]

We observe that [tex]\(x^2 + x\)[/tex] simplifies to [tex]\(x^2 + x\)[/tex], while [tex]\(x(2x + 1)\)[/tex] simplifies to [tex]\(2x^2 + x\)[/tex]. Clearly, [tex]\(x^2 + x\)[/tex] and [tex]\(2x^2 + x\)[/tex] are not identical expressions.

Hence, [tex]\(x^2 + x\)[/tex] and [tex]\(x(2x + 1)\)[/tex] are not equivalent expressions overall, even though they give the same value for [tex]\(x = 0\)[/tex].

Therefore, Kari is incorrect because two expressions that are equivalent must have the same value for all values of [tex]\(x\)[/tex], not just for a single value.