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If a!= 2, which of the following is equa...

If `a!= 2`, which of the following is equal to `(b(a^2 - 4))/(ab - 2b)` ?

A

`ab`

B

`a`

C

`a + 2`

D

`a^2`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the expression \((b(a^2 - 4))/(ab - 2b)\), we will simplify it step by step. ### Step 1: Factor the numerator The numerator is \(b(a^2 - 4)\). We recognize that \(a^2 - 4\) is a difference of squares, which can be factored as: \[ a^2 - 4 = (a - 2)(a + 2) \] Thus, the numerator becomes: \[ b(a^2 - 4) = b((a - 2)(a + 2)) = b(a - 2)(a + 2) \] ### Step 2: Factor the denominator The denominator is \(ab - 2b\). We can factor out \(b\): \[ ab - 2b = b(a - 2) \] ### Step 3: Rewrite the expression Now we can rewrite the entire expression using the factored forms: \[ \frac{b(a - 2)(a + 2)}{b(a - 2)} \] ### Step 4: Cancel common factors Since \(b\) and \(a - 2\) are common in both the numerator and denominator, we can cancel them out (given that \(a \neq 2\) and \(b \neq 0\)): \[ \frac{(a - 2)(a + 2)}{(a - 2)} = a + 2 \] ### Final Result Thus, the expression simplifies to: \[ a + 2 \] ### Conclusion The final answer is: \[ \boxed{a + 2} \]
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