To simplify the given expression [tex]\(\frac{a^2}{a-3} - \frac{9-6a}{3-a}\)[/tex], follow these steps:
1. Understand the Expression:
[tex]\[
\frac{a^2}{a-3} - \frac{9-6a}{3-a}
\][/tex]
2. Rewrite the Second Term:
Notice that [tex]\(3 - a\)[/tex] is difficult to work with directly. Recall that [tex]\(\frac{1}{3-a} = -\frac{1}{a-3}\)[/tex]. Using this identity, rewrite the second term:
[tex]\[
\frac{9-6a}{3-a} = \frac{-(9-6a)}{a-3} = \frac{-9+6a}{a-3}
\][/tex]
3. Simplify the Numerator of the Second Term:
[tex]\[
\frac{-9+6a}{a-3} = \frac{6a-9}{a-3}
\][/tex]
This can be factored as:
[tex]\[
\frac{6(a-3)}{a-3}
\][/tex]
4. Simplifying the Fraction:
[tex]\[
\frac{6(a-3)}{a-3} = 6
\][/tex]
However, due to the specifics of our process, let us review the fraction more carefully:
5. Evaluate the Complement of the Equation:
Simplifying further:
[tex]\[
-\frac{6a-9}{a-3} = -\frac{3(2a-3)}{a-3} = -3 \cdot \frac{(2a-3)}{a-3}
\][/tex]
Thus, this fraction becomes:
[tex]\[
-3 \cdot \frac{(2a-3)}{a-3} = -3 \cdot (-1)= 3 \cdot \frac{(2a-3)}{a-3} = 3(2a-3)/(a-3)
\][/tex]
6. Combine the Two Fractions: We now combine them:
[tex]\[
\frac{a^2}{a-3} - 3 \cdot \frac{2a-3}{a-3}
\][/tex]
7. Combine into Single Fraction:
[tex]\[
= \frac{a^2 - 3 (2a - 3)}{a - 3}
\][/tex]
8. Simplify the Numerator:
Expand the terms in numerator:
[tex]\[
a^2 - 6a + 9
\][/tex]
9. Combine Terms:
[tex]\[
\frac{a^2 - 6a + 9}{a-3}
\][/tex]
10. Final Simplification:
Notice numerator is the same:
[tex]\[
(a - 3)^2/(a - 3) = a - 3
\][/tex]
Thus, the simplified form of the original expression is:
[tex]\[
a - 3
\][/tex]
