Question

\frac{a}{a-5}-\frac{a^2}{a^2-10a+25}

Answer 1

$\frac{a}{a-5}-\frac{a^2}{a^2-10a+25} = \\ = \frac{a}{a-5}-\frac{a^2}{(a - 5)^2} = \frac{a(a - 5) - {a}^{2} }{(a - 5)^2} = \\ = \frac{ {a}^{2} - 5a - {a}^{2} }{ {(a - 5)}^{2} } = \frac{ - 5a }{ {(a - 5)}^{2} }$

Answer 2

To simplify the given expression, we need to find a common denominator for both fractions and then combine them. Let's proceed step by step:

1. Factorize the denominators: The denominator of the first fraction is already in its simplest form: (a - 5). The denominator of the second fraction is a quadratic expression, and it can be factored as follows: a^2 - 10a + 25 = (a - 5)(a - 5)

2. Write the fractions with the common denominator: The common denominator for both fractions is (a - 5)(a - 5).

3. Rewrite the numerators with the common denominator: For the first fraction, we need to multiply the numerator by (a - 5)/(a - 5): \frac{a}{a - 5} = \frac{a \cdot (a - 5)}{(a - 5)(a - 5)} = \frac{a^2 - 5a}{(a - 5)(a - 5)}

   For the second fraction, we need to multiply the numerator by (a - 5)/(a - 5)(a - 5): \frac{a^2}{a^2 - 10a + 25} = \frac{a^2}{(a - 5)(a - 5)} = \frac{a^2}{(a - 5)^2}

4. Combine the fractions: \frac{a}{a - 5} - \frac{a^2}{a^2 - 10a + 25} = \frac{a^2 - 5a}{(a - 5)(a - 5)} - \frac{a^2}{(a - 5)^2}

5. Find a common denominator again (since the current denominators are the same): The common denominator is (a - 5)(a - 5).

6. Combine the numerators: \frac{a^2 - 5a}{(a - 5)(a - 5)} - \frac{a^2}{(a - 5)^2} = \frac{(a^2 - 5a) - a^2}{(a - 5)^2}

7. Simplify the numerator: (a^2 - 5a) - a^2 = -5a

8. Final result: \frac{a}{a - 5} - \frac{a^2}{a^2 - 10a + 25} = \frac{-5a}{(a - 5)^2}

So, the simplified expression is \frac{-5a}{(a - 5)^2}.