Question

Ginger was asked to solve 25x^2−196=0 by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)
Responses

x=19625
x equals Start Fraction 196 over 25 End Fraction

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

There is no solution for x because the difference of squares pattern does not apply.

Answers

Answered by GPT-4o mini
To solve the equation \( 25x^2 - 196 = 0 \) using the difference of squares pattern, we can first recognize that it can be expressed in the form \( a^2 - b^2 \).

1. Rewrite the equation:
\[
25x^2 - 196 = 0
\]

2. Identify \( a \) and \( b \):
- \( a^2 = 25x^2 \) implies \( a = 5x \)
- \( b^2 = 196 \) implies \( b = 14 \)

3. Now, we can use the difference of squares:
\[
(5x)^2 - 14^2 = 0
\]

4. Factoring this gives us:
\[
(5x - 14)(5x + 14) = 0
\]

5. Setting each factor to zero:
\[
5x - 14 = 0 \quad \Rightarrow \quad 5x = 14 \quad \Rightarrow \quad x = \frac{14}{5}
\]

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

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

Thus, the correct response is:
**x = -145, x = 145 (x equals negative Start Fraction 14 over 5 End Fraction, x equals Start Fraction 14 over 5 End Fraction)**.
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