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 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=−145, x=145
x equals negative Start Fraction 14 over 5 End Fraction , x equals Start Fraction 14 over 5 End Fraction

x=196/25

Answer 1

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

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

Now we can use the difference of squares formula \( a^2 - b^2 = (a - b)(a + b) \):

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

Next, we set each factor equal to zero:

1. \( 5x - 14 = 0 \)
2. \( 5x + 14 = 0 \)

Solving the first equation:

\[ 5x - 14 = 0 \implies 5x = 14 \implies x = \frac{14}{5} \]

Solving the second equation:

\[ 5x + 14 = 0 \implies 5x = -14 \implies x = -\frac{14}{5} \]

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 response options provided, the correct solution(s) are:

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