To solve the equation \( 25x^2 - 196 = 0 \) using the difference of squares pattern, we can rewrite the expression in the following way:
The equation is in the form \( a^2 - b^2 = 0 \), where:
- \( a^2 = 25x^2 \) implying \( a = 5x \)
- \( b^2 = 196 \) implying \( b = 14 \)
Now we can factor the left side using the difference of squares formula, which states that \( a^2 - b^2 = (a - b)(a + b) \):
\[ 25x^2 - 196 = (5x - 14)(5x + 14) = 0 \]
Now, we can set each factor equal to zero:
-
\( 5x - 14 = 0 \)
Solving for \( x \), we add 14 to both sides:
\[ 5x = 14 \] \[ x = \frac{14}{5} \]
-
\( 5x + 14 = 0 \)
Solving for \( x \), we subtract 14 from both sides:
\[ 5x = -14 \] \[ x = -\frac{14}{5} \]
Thus, the correct solutions to the equation \( 25x^2 - 196 = 0 \) are:
\[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]
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