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

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

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

x=145

Answer 1

To solve the equation \( 25x^2 - 196 = 0 \) using the difference of squares pattern, we can follow these steps:

1. Recognize that the equation can be rearranged as follows: \[ 25x^2 = 196 \]

2. We can express the left-hand side as \( (5x)^2 \) and realize that \( 196 \) can be expressed as \( 14^2 \). Thus, we can write: \[ (5x)^2 - 14^2 = 0 \]

3. Now we can factor this using the difference of squares: \[ (5x - 14)(5x + 14) = 0 \]

4. Set each factor to zero and solve for \( x \): \[ 5x - 14 = 0 \quad \text{or} \quad 5x + 14 = 0 \]

   From \( 5x - 14 = 0 \): \[ 5x = 14 \implies x = \frac{14}{5} \]

   From \( 5x + 14 = 0 \): \[ 5x = -14 \implies x = -\frac{14}{5} \]

Therefore, the solutions are: \[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]

Considering the responses given in your question, the correct response is: "x equals negative Start Fraction 14 over 5 End Fraction , x equals Start Fraction 14 over 5 End Fraction."