Find the proper subsets of `{x: x^2 -9x -10 = 0}`.

Answer

To find the proper subsets of the set defined by the equation \( x^2 - 9x - 10 = 0 \), we will follow these steps:

### Step 1: Solve the quadratic equation
We start with the equation:
\[ x^2 - 9x - 10 = 0 \]

To factor this quadratic, we look for two numbers that multiply to \(-10\) (the constant term) and add up to \(-9\) (the coefficient of \(x\)). The numbers \(-10\) and \(1\) satisfy these conditions.

### Step 2: Factor the equation
We can rewrite the quadratic as:
\[ (x - 10)(x + 1) = 0 \]

### Step 3: Find the roots
Setting each factor to zero gives us the roots:
\[ x - 10 = 0 \quad \Rightarrow \quad x = 10 \]
\[ x + 1 = 0 \quad \Rightarrow \quad x = -1 \]

### Step 4: Define the set
Thus, the set \( A \) is:
\[ A = \{-10, 1\} \]

### Step 5: Find the proper subsets
The proper subsets of a set are all the subsets excluding the set itself. For the set \( A = \{-10, 1\} \), the proper subsets are:
1. The empty set: \( \emptyset \)
2. The set containing just \(-10\): \( \{-10\} \)
3. The set containing just \(1\): \( \{1\} \)

### Final Result
The proper subsets of the set \( A \) are:
\[ \emptyset, \{-10\}, \{1\} \]