To solve the equation \( 25x^2 - 196 = 0 \) by factoring, we can start by recognizing that it is a difference of squares. The equation can be rewritten as:
\[ 25x^2 - 196 = (5x)^2 - (14)^2 = 0 \]
Now we can factor it using the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \):
\[ (5x - 14)(5x + 14) = 0 \]
Next, we set each factor equal 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:
\[ 5x + 14 = 0 \implies 5x = -14 \implies x = -\frac{14}{5} \]
Thus, the solutions to the equation \( 25x^2 - 196 = 0 \) are:
\[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]
None of the provided response options (including \( x = 19625 \), \( x = 145 \), etc.) seem to match these correct solutions. The right answers should be:
\[ x = \frac{14}{5}, , x = -\frac{14}{5} \]