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)

Question

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=145
x equals Start Fraction 14 over 5 End Fraction

x=19625
x equals Start Fraction 196 over 25 End Fraction

There is no solution for x because the difference of squares pattern does not apply.

Answer 1

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

\[ (5x)^2 - 14^2 = 0 \]

This is in the form \( a^2 - b^2 = 0 \), which factors to:

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

Setting each factor to zero gives us:

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:

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

Among the options provided, the correct response is:

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