To solve the equation \( 25x^2 - 196 = 0 \) by factoring using the difference of squares pattern, we first recognize that both \( 25x^2 \) and \( 196 \) can be rewritten as squares:
\[ 25x^2 = (5x)^2 \] \[ 196 = 14^2 \]
This allows us to rewrite the equation as:
\[ (5x)^2 - 14^2 = 0 \]
Now we can apply the difference of squares factoring, which states that \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = 5x \) and \( b = 14 \), so we have:
\[ (5x - 14)(5x + 14) = 0 \]
Next, we set each factor equal to zero:
- \( 5x - 14 = 0 \)
- \( 5x + 14 = 0 \)
Now, we can solve for \( x \) in each case:
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For \( 5x - 14 = 0 \): \[ 5x = 14 \quad \Rightarrow \quad x = \frac{14}{5} \]
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For \( 5x + 14 = 0 \): \[ 5x = -14 \quad \Rightarrow \quad x = -\frac{14}{5} \]
Thus, the solutions to the equation \( 25x^2 - 196 = 0 \) are:
\[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]
In conclusion, the correct solutions are: \[ \boxed{\frac{14}{5}, -\frac{14}{5}} \]