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

A) x=14/5
x equals Start Fraction 14 over 5 End Fraction

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

C) x=−14/5, x=14/5
x equals negative Start Fraction 14 over 5 End Fraction , x equals Start Fraction 14 over 5 End Fraction

D) x=196/25

Answer 1

To solve the equation \( 25x^2 - 196 = 0 \) using the difference of squares pattern, we first recognize that this can be rewritten in the form \( a^2 - b^2 = 0 \).

Notice that:

\[ 25x^2 = (5x)^2 \quad \text{and} \quad 196 = 14^2 \]

So we can rewrite the equation as:

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

This can be factored using the difference of squares formula:

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

Thus, we have:

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

Setting each factor equal to zero gives us:

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

   • Solving for \( x \) gives \( 5x = 14 \) or \( x = \frac{14}{5} \).

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

   • Solving for \( x \) gives \( 5x = -14 \) or \( x = -\frac{14}{5} \).

Therefore, the solutions to the equation are:

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

The correct answer is: C) \( x = -\frac{14}{5} \), \( x = \frac{14}{5} \).