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

Question

A. x = - 14/5, x = 14/5
B. x = 14/5
C. x = 196/25
D. there is no solution for x because the difference of squares pattern does not apply.

Answer 1

To solve the equation \( 25x^2 - 196 = 0 \) using the difference of squares pattern, we can first rewrite it in the form of a difference of squares.

The equation can be expressed as:

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

This fits the difference of squares formula, \( a^2 - b^2 = (a - b)(a + b) \), where \( a = 5x \) and \( b = 14 \).

Using the difference of squares, we factor the equation:

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

Now, set each factor equal to zero:

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:

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

Therefore, the correct answer is:

A. \( x = -\frac{14}{5}, x = \frac{14}{5} \).