Given:
[tex]$\frac{6}{a^{2}-7 a+6}, \frac{3}{a^{2}-36}[/tex]
To find:
The LCD of the fractions.
Solution:
LCD means least common denominator.
Let us find the least common multiplier for the denominator.
The denominators are [tex]\left(a^{2}-7 a+a\right),\left(a^{2}-36\right)[/tex].
Factor [tex]\left(a^{2}-7 a+a\right)[/tex]:
[tex]a^{2}-7 a+6=\left(a^{2}-a\right)+(-6 a+6)[/tex]
Take a common in first 2 terms and -6 common in next two terms.
[tex]=a(a-1)-6(a-1)[/tex]
Take out common factor (a - 1).
[tex]\left(a^{2}-7 a+a\right)=(a-1)(a-6)[/tex] ------------- (1)
Factor [tex]\left(a^{2}-36\right)[/tex]:
[tex]\left(a^{2}-36\right)=\left(a^{2}-6^2\right)[/tex]
Using identity: [tex](a^2-b^2)=(a-b)(a+b)[/tex]
[tex]\left(a^{2}-36\right)=(a-6)(a+6)[/tex] ------------- (1)
From (1) and (2),
LCM of [tex]\left(a^{2}-7 a+a\right),\left(a^{2}-36\right)=(a-1)(a-6)(a+6)[/tex]
Therefore LCD is (a - 1)(a - 6)(a + 6).
