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

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=145
x equals Start Fraction 14 over 5 End Fraction

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

Answer 1

To solve the equation \(25x^2 - 196 = 0\) by factoring using the difference of squares pattern, we can start by recognizing that this expression can be rearranged and factored.

1. The equation can be rewritten as: \[ 25x^2 = 196 \]

2. Next, we can recognize that both sides can be expressed as squares: \[ (5x)^2 = 14^2 \]

3. Now, we can apply the difference of squares formula, which states that \(a^2 - b^2 = (a - b)(a + b)\). This gives us: \[ (5x - 14)(5x + 14) = 0 \]

4. Setting each factor equal to zero gives: \[ 5x - 14 = 0 \quad \text{or} \quad 5x + 14 = 0 \]

5. Solving these equations:

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

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

Given the options:

• \(x=-\frac{14}{5}, ; x=\frac{14}{5}\)

This is the correct response.