Ginger was asked to solve 25x^2−196=0

Question

by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
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=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 rewrite it in a suitable form.

The equation can be recognized as a difference of squares: \[ 25x^2 - 196 = (5x)^2 - (14)^2 \] This follows the pattern \( a^2 - b^2 = (a - b)(a + b) \). Here, \( a = 5x \) and \( b = 14 \).

Using the difference of squares factoring, we have: \[ (5x - 14)(5x + 14) = 0 \]

Now, we can set each factor to zero and solve for \( x \):

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

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

Thus, the solutions to the equation \( 25x^2 - 196 = 0 \) are: \[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]

Looking at the given options, the correct response is:

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