Questions: (y^2+y-14)/(y^2-49)+(2/(y+7))

Transcript text: 1. $\frac{y^{2}+y-14}{y^{2}-49}+\frac{2}{y+7}$

Solution

Solution Steps

Step 1: Factor the Denominator

The denominator of the expression $ \frac{y^{2}+y-14}{y^{2}-49}+\frac{2}{y+7} $ is $ y^{2} - 49 $. This can be factored as: \[ y^{2} - 49 = (y - 7)(y + 7) \] Thus, the factorized form of the denominator is $ (y - 7)(y + 7) $.

Step 2: Combine the Fractions

We can rewrite the expression using the factorized form: \[ \frac{y^{2} + y - 14}{(y - 7)(y + 7)} + \frac{2}{y + 7} \] To combine these fractions, we need a common denominator: \[ \frac{y^{2} + y - 14 + 2(y - 7)}{(y - 7)(y + 7)} \] Simplifying the numerator: \[ y^{2} + y - 14 + 2y - 14 = y^{2} + 3y - 28 \] Thus, the combined expression is: \[ \frac{y^{2} + 3y - 28}{(y - 7)(y + 7)} \]

Step 3: Perform Partial Fraction Decomposition

Next, we perform partial fraction decomposition on the expression: \[ \frac{y^{2} + 3y - 28}{(y - 7)(y + 7)} \] This can be expressed as: \[ \frac{A}{y - 7} + \frac{B}{y + 7} \] After performing the decomposition, we find: \[ \frac{y - 4}{y - 7} \]

Final Answer

The final result of the expression $ \frac{y^{2}+y-14}{y^{2}-49}+\frac{2}{y+7} $ after simplification and partial fraction decomposition is: \[ \boxed{\frac{y - 4}{y - 7}} \]

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