State, whether the following pairs of sets are equal or not:
`E={x:x^(2) + 8x-9=0}` and `F={1,-9}`

Answer

To determine whether the sets \( E \) and \( F \) are equal, we need to follow these steps:

### Step 1: Identify the sets
We have:
- Set \( E = \{ x : x^2 + 8x - 9 = 0 \} \)
- Set \( F = \{ 1, -9 \} \)

### Step 2: Solve the equation for set \( E \)
To find the elements of set \( E \), we need to solve the quadratic equation \( x^2 + 8x - 9 = 0 \).

### Step 3: Factor the quadratic equation
We can factor the equation as follows:
1. We look for two numbers that multiply to \(-9\) (the constant term) and add to \(8\) (the coefficient of \(x\)).
2. The numbers \(9\) and \(-1\) satisfy this condition.

Thus, we can rewrite the equation:
\[ x^2 + 9x - 1x - 9 = 0 \]
Grouping the terms:
\[ x(x + 9) - 1(x + 9) = 0 \]
Factoring by grouping:
\[ (x - 1)(x + 9) = 0 \]

### Step 4: Find the roots
Setting each factor to zero gives us:
1. \( x - 1 = 0 \) → \( x = 1 \)
2. \( x + 9 = 0 \) → \( x = -9 \)

Thus, the solutions to the equation are \( x = 1 \) and \( x = -9 \).

### Step 5: Write the set \( E \)
Now we can express set \( E \):
\[ E = \{ 1, -9 \} \]

### Step 6: Compare sets \( E \) and \( F \)
Now we compare the elements of both sets:
- Set \( E = \{ 1, -9 \} \)
- Set \( F = \{ 1, -9 \} \)

### Step 7: Determine if the sets are equal
Since both sets contain the same elements, we conclude that:
\[ E = F \]

### Final Answer
The sets \( E \) and \( F \) are equal.

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