Factor the following quadratic expressions.
`x^(2)-14x+45`

Answer

To factor the quadratic expression \( x^2 - 14x + 45 \), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the standard form \( ax^2 + bx + c \). Here, we have:
- \( a = 1 \)
- \( b = -14 \)
- \( c = 45 \)

### Step 2: Find two numbers that multiply to \( ac \) and add to \( b \)
We need to find two numbers that multiply to \( ac = 1 \times 45 = 45 \) and add up to \( b = -14 \).

The two numbers that satisfy these conditions are:
- \( -9 \) and \( -5 \) (since \( -9 \times -5 = 45 \) and \( -9 + -5 = -14 \))

### Step 3: Rewrite the middle term using the two numbers
We can rewrite the expression \( x^2 - 14x + 45 \) as:
\[ x^2 - 9x - 5x + 45 \]

### Step 4: Factor by grouping
Now, we will group the terms:
\[ (x^2 - 9x) + (-5x + 45) \]

Next, we factor out the common factors from each group:
- From the first group \( x^2 - 9x \), we can factor out \( x \):
\[ x(x - 9) \]
- From the second group \( -5x + 45 \), we can factor out \( -5 \):
\[ -5(x - 9) \]

Now, we can combine these factored forms:
\[ x(x - 9) - 5(x - 9) \]

### Step 5: Factor out the common binomial factor
Now, we see that \( (x - 9) \) is a common factor:
\[ (x - 9)(x - 5) \]

### Final Answer
Thus, the expression \( x^2 - 14x + 45 \) factors to:
\[ (x - 5)(x - 9) \]

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