To solve the equation \( 25x^2 - 196 = 0 \) by factoring using the difference of squares pattern, we can recognize that \( 25x^2 \) and \( 196 \) are both perfect squares.
The difference of squares can be factored as follows:
\[ a^2 - b^2 = (a - b)(a + b) \]
In this case, we have:
\[ a = 5x \quad \text{(since } (5x)^2 = 25x^2\text{)} \] \[ b = 14 \quad \text{(since } 14^2 = 196\text{)} \]
So we can rewrite the equation:
\[ 25x^2 - 196 = (5x - 14)(5x + 14) = 0 \]
Setting each factor to zero gives:
- \( 5x - 14 = 0 \)
- \( 5x + 14 = 0 \)
Solving the first equation:
\[ 5x - 14 = 0 \] \[ 5x = 14 \] \[ x = \frac{14}{5} \]
Solving the second equation:
\[ 5x + 14 = 0 \] \[ 5x = -14 \] \[ x = -\frac{14}{5} \]
Therefore, the correct solutions are:
\[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]
From the provided options, the correct response corresponds with:
\( x = -\frac{14}{5} \) and \( x = \frac{14}{5} \) (but both are actually the same value, just with different signs).
To summarize, the complete solution set is:
\( x = \frac{14}{5} \) and \( x = -\frac{14}{5} \).
1 answer