Ginger was asked to solve 25x2−196=0 by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point) Responses x=−145, x=145 x equals negative Start Fraction 14 over 5 End Fraction , x equals Start Fraction 14 over 5 End Fraction x=19625 x equals Start Fraction 196 over 25 End Fraction x=145 x equals Start Fraction 14 over 5 End Fraction There is no solution for x because the difference of squares pattern does not apply.

Answer 1

To solve the equation \( 25x^2 - 196 = 0 \) by factoring using the difference of squares pattern, we can recognize that this can be expressed as a difference of squares:

\[ 25x^2 - 196 = (5x)^2 - (14)^2 \]

Using the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \), we can factor this:

\[ (5x - 14)(5x + 14) = 0 \]

Now, we will set each factor equal to zero:

1. \( 5x - 14 = 0 \) \[ 5x = 14 \quad \Rightarrow \quad x = \frac{14}{5} \]

2. \( 5x + 14 = 0 \) \[ 5x = -14 \quad \Rightarrow \quad x = -\frac{14}{5} \]

Thus, the solutions to the equation \( 25x^2 - 196 = 0 \) are:

\[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]

So the correct response is:

x equals negative Start Fraction 14 over 5 End Fraction , x equals Start Fraction 14 over 5 End Fraction.