To find the Highest Common Factor (HCF) of the given algebraic expressions \( x^2 - 5x + 6 \) and \( x^2 - 9 \), follow these steps:
1. Factorize Each Expression:
- For \( x^2 - 5x + 6 \), we look for factors of 6 that add up to -5. Those factors are -2 and -3, so we can write:
[tex]\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\][/tex]
- For \( x^2 - 9 \), recognize that this is a difference of squares. We can write:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
2. Identify Common Factors:
- Both factorizations contain the factor \( x - 3 \):
[tex]\[
x^2 - 5x + 6 = (x - 2)(x - 3)
\][/tex]
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
3. Determine the HCF:
The common factor between the two factorizations is \( x - 3 \).
Therefore, the HCF of the expressions \( x^2 - 5x + 6 \) and \( x^2 - 9 \) is:
[tex]\[
x - 3
\][/tex]
