Question

add the following fractions. simplify your answer as much as possible.\\(\frac{6z^{2}-11z + 14}{z^{2}-4z - 12}-\frac{5z^{2}-3z + 2}{z^{2}-4z - 12}\\)\
answer\\((z + 6)/(z - 6)\\)\
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check answer attempt 3 of 3

Explanation:

Step1: Subtract numerators (common denominator)

Since the denominators are the same ($z^2 - 4z - 12$), we subtract the numerators:
$(6z^2 - 11z + 14) - (5z^2 - 3z + 2)$

Step2: Simplify the numerator

Expand and combine like terms:
$6z^2 - 11z + 14 - 5z^2 + 3z - 2 = (6z^2 - 5z^2) + (-11z + 3z) + (14 - 2) = z^2 - 8z + 12$

Step3: Factor numerator and denominator

Factor $z^2 - 8z + 12$: $z^2 - 8z + 12 = (z - 2)(z - 6)$
Factor $z^2 - 4z - 12$: $z^2 - 4z - 12 = (z - 6)(z + 2)$

Step4: Cancel common factors

The fraction becomes $\frac{(z - 2)(z - 6)}{(z - 6)(z + 2)}$. Cancel $(z - 6)$ (assuming $z
eq 6$):
$\frac{z - 2}{z + 2}$

Answer:

$\frac{z - 2}{z + 2}$