Question

Solve the equation by factoring. What is/ are the correct solution(s)?

25x^2−196=0
(1 point)
Responses

x=−145, x=145
x=−145, x=145

No solution
No solution

x=19625
x is equal to 196 over 25

x=145

Answer 1

To solve the equation \( 25x^2 - 196 = 0 \) by factoring, we can start by recognizing that it is a difference of squares. The equation can be rewritten as:

\[
25x^2 - 196 = (5x)^2 - (14)^2 = 0
\]

Now we can factor it using 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:

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

None of the provided response options (including \( x = 19625 \), \( x = 145 \), etc.) seem to match these correct solutions. The right answers should be:

\[
x = \frac{14}{5}, \, x = -\frac{14}{5}
\]