To simplify the expression \( \frac{20m^2 - 5}{8m - 4} \), we can start by factoring both the numerator and the denominator.
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Factor the numerator: \[ 20m^2 - 5 = 5(4m^2 - 1) \] We can recognize \( 4m^2 - 1 \) as a difference of squares: \[ 4m^2 - 1 = (2m - 1)(2m + 1) \] Thus, the numerator can be factored as: \[ 20m^2 - 5 = 5(2m - 1)(2m + 1) \]
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Factor the denominator: \[ 8m - 4 = 4(2m - 1) \]
Now we can rewrite the original expression using the factored forms:
\[ \frac{20m^2 - 5}{8m - 4} = \frac{5(2m - 1)(2m + 1)}{4(2m - 1)} \]
- Cancel the common factor \( (2m - 1) \) in the numerator and denominator (provided \( 2m - 1 \neq 0 \) or \( m \neq \frac{1}{2} \)): \[ \frac{5(2m + 1)}{4} \]
So, the simplified expression is:
\[ \frac{5(2m + 1)}{4} \quad \text{or} \quad \frac{10m + 5}{4} \text{ if you expand it again.} \]
Final simplified form: \[ \frac{5(2m + 1)}{4} \]