To factor the expression [tex]\( x^2 - 14x + 45 \)[/tex] completely, we'll follow these steps:
1. Identify the quadratic expression: The given expression is [tex]\( x^2 - 14x + 45 \)[/tex].
2. Set up the quadratic equation: Since we are factoring, we look for two numbers that multiply to the constant term (45) and add up to the coefficient of the linear term (-14).
3. Find the numbers: We need to find two numbers [tex]\( a \)[/tex] and [tex]\( b \)[/tex] such that:
[tex]\[
a \cdot b = 45 \quad \text{and} \quad a + b = -14
\][/tex]
4. Possible pairs for 45: We consider pairs of factors of 45:
- [tex]\( 1 \cdot 45 \)[/tex]
- [tex]\( 3 \cdot 15 \)[/tex]
- [tex]\( 5 \cdot 9 \)[/tex]
5. Determine the correct pair: We need the pair that sums to -14. Checking the factor pairs:
- [tex]\( 1 + 45 = 46 \)[/tex] (not correct)
- [tex]\( 3 + 15 = 18 \)[/tex] (not correct)
- [tex]\( 5 + 9 = 14 \)[/tex] (when considering the signs correctly, if both are negative: [tex]\( (-5) + (-9) = -14 \)[/tex])
The correct pair is [tex]\( -5 \)[/tex] and [tex]\( -9 \)[/tex].
6. Factor the quadratic expression: Given these numbers, we can write the quadratic expression as:
[tex]\[
x^2 - 14x + 45 = (x - 5)(x - 9)
\][/tex]
Thus, the factored form of [tex]\( x^2 - 14x + 45 \)[/tex] is:
[tex]\[
(x - 5)(x - 9)
\][/tex]
