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Question
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)

Answer 1

To solve the equation \( 25x^2 - 196 = 0 \) by factoring using the difference of squares pattern, we can rewrite it in the form \( a^2 - b^2 = 0 \), where \( a = 5x \) and \( b = 14 \) because \( 25x^2 = (5x)^2 \) and \( 196 = 14^2 \).

So, we can write:

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

Now we can apply the difference of squares formula:

\[
a^2 - b^2 = (a - b)(a + b)
\]

This gives us:

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

Now we can set each factor 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:

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

Therefore, the solutions are:

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