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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)
Responses

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

x=145
x equals Start Fraction 14 over 5 End Fraction

x=19625

1 answer

To solve the equation \( 25x^2 - 196 = 0 \) using the difference of squares pattern, we can first recognize that it can be rewritten as:

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

This fits the difference of squares form, \( a^2 - b^2 = 0 \), where \( a = 5x \) and \( b = 14 \). The difference of squares can be factored as follows:

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

Now, we set each factor equal to zero:

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

    • Adding 14 to both sides: \[ 5x = 14 \implies x = \frac{14}{5} \]

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

    • Subtracting 14 from both sides: \[ 5x = -14 \implies x = -\frac{14}{5} \]

Thus, the solutions are:

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

Hence, the correct response from the given options is:

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