Questions: (y+7)/(y^2+2y-15)-(4)/(y^2-25)

Solution Steps

To simplify the given expression, we need to perform the subtraction of two rational expressions. The first step is to factor the denominators of both fractions. Once factored, we find a common denominator, rewrite each fraction with this common denominator, and then perform the subtraction. Finally, we simplify the resulting expression if possible.

We start by factoring the denominators of the two fractions: \[ y^2 + 2y - 15 = (y - 3)(y + 5) \] \[ y^2 - 25 = (y - 5)(y + 5) \]

The common denominator for the two fractions is: \[ (y - 5)(y - 3)(y + 5)^2 \]

We rewrite each fraction with the common denominator: \[ \frac{y + 7}{(y - 3)(y + 5)} = \frac{(y + 7)(y - 5)}{(y - 5)(y - 3)(y + 5)^2} \] \[ \frac{4}{(y - 5)(y + 5)} = \frac{4(y - 3)(y + 5)}{(y - 5)(y - 3)(y + 5)^2} \]

Now we perform the subtraction: \[ \frac{(y + 7)(y - 5) - 4(y - 3)(y + 5)}{(y - 5)(y - 3)(y + 5)^2} \]

Expanding and simplifying the numerator: \[ (y + 7)(y - 5) = y^2 + 2y - 35 \] \[ 4(y - 3)(y + 5) = 4(y^2 + 2y - 15) = 4y^2 + 8y - 60 \] Thus, the numerator becomes: \[ y^2 + 2y - 35 - (4y^2 + 8y - 60) = -3y^2 - 6y + 25 \] So, the simplified expression is: \[ \frac{-3y^2 - 6y + 25}{(y - 5)(y - 3)(y + 5)^2} \]

Final Answer