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=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 x=19625 x equals Start Fraction 196 over 25 End Fraction There is no solution for x because the difference of squares pattern does not apply. 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 first recognize that this can be factored as:

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

This matches the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \).

So we can rewrite the equation:

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

Next, we set each factor equal to zero:

1. \( 5x - 14 = 0 \): \[ 5x = 14 \quad \Rightarrow \quad x = \frac{14}{5} \]

2. \( 5x + 14 = 0 \): \[ 5x = -14 \quad \Rightarrow \quad x = -\frac{14}{5} \]

Thus, the solutions are: \[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]

The corresponding response would be:

x equals negative Start Fraction 14 over 5 End Fraction , x equals Start Fraction 14 over 5 End Fraction.