To factor the expression \( 256q^2 - 196 \), we can first recognize that this is a difference of squares. The expression can be rewritten as:
\[ (16q)^2 - 14^2 \]
Using the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \), we can factor the expression:
Let \( a = 16q \) and \( b = 14 \). Therefore:
\[ 256q^2 - 196 = (16q - 14)(16q + 14) \]
Next, we can simplify the factors. Both \( 16q - 14 \) and \( 16q + 14 \) can be factored further.
Factoring out the common factor of 2 from both terms, we get:
\[ 16q - 14 = 2(8q - 7) \] \[ 16q + 14 = 2(8q + 7) \]
Now we can substitute back into our original factorization:
\[ 256q^2 - 196 = (16q - 14)(16q + 14) = (2(8q - 7))(2(8q + 7)) \]
Combining the constants, we have:
\[ = 4(8q - 7)(8q + 7) \]
Thus, the completely factored form of \( 256q^2 - 196 \) is:
\[ \boxed{4(8q - 7)(8q + 7)} \]
2 answers