List the elements of the following sets:
`x^(2)-2x -3=0`

Answer

To solve the equation \( x^2 - 2x - 3 = 0 \) and list the elements of the set, we will follow these steps:

### Step 1: Write the quadratic equation
The given equation is:
\[ x^2 - 2x - 3 = 0 \]

### Step 2: Factor the quadratic equation
To factor the quadratic equation, we look for two numbers that multiply to \(-3\) (the constant term) and add to \(-2\) (the coefficient of \(x\)). The numbers \(-3\) and \(1\) satisfy these conditions.

We can rewrite the equation as:
\[ x^2 - 3x + x - 3 = 0 \]

### Step 3: Group the terms
Now, we can group the terms:
\[ (x^2 - 3x) + (x - 3) = 0 \]

### Step 4: Factor by grouping
Next, we factor out the common terms in each group:
\[ x(x - 3) + 1(x - 3) = 0 \]

Now we can factor out \( (x - 3) \):
\[ (x - 3)(x + 1) = 0 \]

### Step 5: Set each factor to zero
Now, we set each factor equal to zero:
1. \( x - 3 = 0 \) → \( x = 3 \)
2. \( x + 1 = 0 \) → \( x = -1 \)

### Step 6: List the elements of the set
The solutions to the equation are the elements of the set:
\[ \{ -1, 3 \} \]

### Final Answer
The elements of the set are:
\[ \{-1, 3\} \]

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