Find the roots of `x^(2) - 32x -900= 0`.

Answer

To find the roots of the quadratic equation \( x^2 - 32x - 900 = 0 \), we can use the method of factoring. Here’s a step-by-step solution:

### Step 1: Write down the equation
We start with the equation:
\[ x^2 - 32x - 900 = 0 \]

### Step 2: Identify coefficients
In the equation \( ax^2 + bx + c = 0 \), we have:
- \( a = 1 \)
- \( b = -32 \)
- \( c = -900 \)

### Step 3: Factor the quadratic
We need to find two numbers that multiply to \( ac = 1 \times (-900) = -900 \) and add to \( b = -32 \).

After checking pairs of factors of -900, we find:
- \( 50 \) and \( -18 \) satisfy both conditions because:
- \( 50 \times (-18) = -900 \)
- \( 50 + (-18) = 32 \)

### Step 4: Rewrite the equation using these factors
We can rewrite the middle term of the equation using these factors:
\[ x^2 - 50x + 18x - 900 = 0 \]

### Step 5: Group the terms
Now, we group the terms:
\[ (x^2 - 50x) + (18x - 900) = 0 \]

### Step 6: Factor by grouping
Factoring out common terms from each group:
\[ x(x - 50) + 18(x - 50) = 0 \]

### Step 7: Factor out the common binomial
Now we can factor out \( (x - 50) \):
\[ (x - 50)(x + 18) = 0 \]

### Step 8: Set each factor to zero
Now, we set each factor equal to zero:
1. \( x - 50 = 0 \) → \( x = 50 \)
2. \( x + 18 = 0 \) → \( x = -18 \)

### Step 9: State the roots
Thus, the roots of the equation are:
\[ x = 50 \quad \text{and} \quad x = -18 \]