Is `{x : x^(2) - 5x -6=0}`={2,3}` ?

Answer

To determine whether the sets `{x : x^2 - 5x - 6 = 0}` and `{2, 3}` are equal, we need to solve the equation and find the elements of the first set.

### Step-by-Step Solution:

1. **Identify the equation**: We start with the equation given in the set notation:
\[ x^2 - 5x - 6 = 0 \]

2. **Factor the quadratic equation**: We need to factor the quadratic expression. We can rewrite it as:
\[ x^2 - 6x + x - 6 = 0 \]
Now, we can group the terms:
\[ (x^2 - 6x) + (x - 6) = 0 \]
Factoring out common terms:
\[ x(x - 6) + 1(x - 6) = 0 \]
This can be factored as:
\[ (x + 1)(x - 6) = 0 \]

3. **Find the roots**: Set each factor to zero to find the values of \(x\):
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]
\[ x - 6 = 0 \quad \Rightarrow \quad x = 6 \]
Thus, the solutions to the equation are:
\[ x = -1 \quad \text{and} \quad x = 6 \]

4. **Form the set**: Therefore, the set defined by the equation is:
\[ \{x : x^2 - 5x - 6 = 0\} = \{-1, 6\} \]

5. **Compare the sets**: Now we compare the two sets:
- Set A: \(\{-1, 6\}\)
- Set B: \(\{2, 3\}\)

We can see that the elements of Set A (-1 and 6) are not in Set B (2 and 3), and the elements of Set B are not in Set A.

6. **Conclusion**: Since the two sets do not contain the same elements, we conclude that:
\[ \{x : x^2 - 5x - 6 = 0\} \neq \{2, 3\} \]

### Final Answer:
No, the sets are not equal.

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