To solve the equation \( 25x^2 - 196 = 0 \) by factoring using the difference of squares pattern, we can rewrite it in the form \( a^2 - b^2 = 0 \), where \( a = 5x \) and \( b = 14 \) because \( 25x^2 = (5x)^2 \) and \( 196 = 14^2 \).
So, we can write:
\[ 25x^2 - 196 = (5x)^2 - 14^2 = 0 \]
Now we can apply the difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
This gives us:
\[ (5x - 14)(5x + 14) = 0 \]
Now we can set each factor to zero:
- \( 5x - 14 = 0 \)
- \( 5x + 14 = 0 \)
Solving the first equation:
\[ 5x - 14 = 0 \implies 5x = 14 \implies x = \frac{14}{5} \]
Solving the second equation:
\[ 5x + 14 = 0 \implies 5x = -14 \implies x = -\frac{14}{5} \]
Therefore, the solutions are:
\[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]
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