(a+2/a-2 - a-2/3+2) :10a/a’2 - 4
To solve the expression \((\frac{a + 2}{a - 2} - \frac{a - 2}{3 + 2}) : \frac{10a}{a^2 - 4}\), let's simplify it step-by-step.
Simplify the first part of the expression:
\[ \frac{a + 2}{a - 2} - \frac{a - 2}{5} \]
To combine these fractions, we need a common denominator. The common denominator will be \(5(a - 2)\).
Hence:
\[ \frac{(a + 2) * 5}{5(a - 2)} - \frac{(a - 2)(a - 2)}{5(a - 2)} \]
Now substitute the equations to combine:
\[ = \frac{5(a + 2) - (a - 2)(a - 2)}{5(a - 2)} \]
Next, we need to expand \( (a - 2)(a - 2) \):
\[ (a - 2)(a - 2) = a^2 - 4a + 4 \]
Now substituting this back:
\[ = \frac{5a + 10 - (a^2 - 4a + 4)}{5(a - 2)} \] \[ = \frac{5a + 10 - a^2 + 4a - 4}{5(a - 2)} \] \[ = \frac{-a^2 + 9a + 6}{5(a - 2)} \]
Rewrite the whole expression:
We now simplify the entire expression:
\[ \frac{-a^2 + 9a + 6}{5(a - 2)} : \frac{10a}{a^2 - 4} \]
The division of fractions can be rewritten as multiplication by the reciprocal:
\[ = \frac{-a^2 + 9a + 6}{5(a - 2)} \cdot \frac{a^2 - 4}{10a} \]
Notice that \(a^2 - 4\) can be factored:
\[ a^2 - 4 = (a - 2)(a + 2) \]
So we rewrite it as:
\[ = \frac{-a^2 + 9a + 6}{5(a - 2)} \cdot \frac{(a - 2)(a + 2)}{10a} \]
Cancel out the common factors:
We see that \((a - 2)\) can be cancelled:
\[ = \frac{-a^2 + 9a + 6}{5 \cdot 10a} \cdot (a + 2) \]
Thus:
\[ = \frac{-a^2 + 9a + 6}{50a} \cdot (a + 2) \]
Finally, let's rewrite the expression:
\[ = \frac{(-a^2 + 9a + 6)(a + 2)}{50a} \]
This step-by-step process helps simplify the given complex expression, and this final form is the simplified result. From here, you can either simplify further based on specific \(a\) values, or convert to polynomial form if needed.
