solve the equation` x^(2) - 18x + 45 = 0 `

Answer

To solve the equation \( x^2 - 18x + 45 = 0 \), we can use the method of middle term factorization. Here’s the step-by-step solution:

### Step 1: Write the equation
We start with the quadratic equation:
\[ x^2 - 18x + 45 = 0 \]

### Step 2: Identify the coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify:
- \( a = 1 \)
- \( b = -18 \)
- \( c = 45 \)

### Step 3: Find two numbers that multiply to \( c \) and add to \( b \)
We need to find two numbers that multiply to \( 45 \) (the constant term) and add up to \( -18 \) (the coefficient of \( x \)).

The numbers that satisfy these conditions are \( -15 \) and \( -3 \):
- \( -15 \times -3 = 45 \)
- \( -15 + (-3) = -18 \)

### Step 4: Rewrite the middle term using these numbers
We can rewrite the equation by splitting the middle term:
\[ x^2 - 15x - 3x + 45 = 0 \]

### Step 5: Factor by grouping
Now, we group the terms:
\[ (x^2 - 15x) + (-3x + 45) = 0 \]
Factoring out the common factors in each group:
\[ x(x - 15) - 3(x - 15) = 0 \]

### Step 6: Factor out the common binomial
Now we can factor out \( (x - 15) \):
\[ (x - 15)(x - 3) = 0 \]

### Step 7: Set each factor to zero
Now, we can set each factor equal to zero:
1. \( x - 15 = 0 \) → \( x = 15 \)
2. \( x - 3 = 0 \) → \( x = 3 \)

### Step 8: Write the final solution
The solutions to the equation \( x^2 - 18x + 45 = 0 \) are:
\[ x = 15 \quad \text{and} \quad x = 3 \]