Solve the equation `x^(2)-45x+324=0`.

Answer

To solve the quadratic equation \( x^2 - 45x + 324 = 0 \), we will use the method of splitting the middle term.

### Step-by-Step Solution:

1. **Identify the coefficients**:
The given equation is in the standard form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = -45 \)
- \( c = 324 \)

2. **Find two numbers that multiply to \( c \) and add up to \( b \)**:
We need to find two numbers that multiply to \( 324 \) (the constant term) and add up to \( -45 \) (the coefficient of \( x \)).
- The factors of \( 324 \) are: \( (1, 324), (2, 162), (3, 108), (4, 81), (6, 54), (9, 36), (12, 27), (18, 18) \).

3. **Select the correct pair**:
We need a pair that adds up to \( -45 \). The pair \( -9 \) and \( -36 \) works because:
- \( -9 \times -36 = 324 \)
- \( -9 + (-36) = -45 \)

4. **Rewrite the equation**:
We can rewrite the middle term \( -45x \) using the pair found:
\[ x^2 - 9x - 36x + 324 = 0 \]

5. **Factor by grouping**:
Group the terms:
\[ (x^2 - 9x) + (-36x + 324) = 0 \]
Factor out the common terms:
\[ x(x - 9) - 36(x - 9) = 0 \]

6. **Factor out the common binomial**:
\[ (x - 9)(x - 36) = 0 \]

7. **Set each factor to zero**:
Now we can set each factor to zero to find the values of \( x \):
- \( x - 9 = 0 \) gives \( x = 9 \)
- \( x - 36 = 0 \) gives \( x = 36 \)

8. **Conclusion**:
The solutions to the equation \( x^2 - 45x + 324 = 0 \) are:
\[ x = 9 \quad \text{and} \quad x = 36 \]