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

x=14/5
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=196/25

1 answer

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

This equation can be expressed as: \[ 25x^2 - 196 = 0 \]

Recognizing that this is a difference of squares, we rewrite it in factored form: \[ (5x)^2 - (14)^2 = 0 \]

Using the difference of squares formula \(a^2 - b^2 = (a - b)(a + b)\), we can factor the equation as follows: \[ (5x - 14)(5x + 14) = 0 \]

Setting each factor equal to zero gives us the possible solutions for \(x\):

  1. \(5x - 14 = 0\)
  2. \(5x + 14 = 0\)

Solving these equations individually:

  1. From \(5x - 14 = 0\): \[ 5x = 14 \implies x = \frac{14}{5} \]
  2. From \(5x + 14 = 0\): \[ 5x = -14 \implies x = -\frac{14}{5} \]

Thus the solutions to the equation \(25x^2 - 196 = 0\) are: \[ x = \frac{14}{5} \quad \text{and} \quad x = -\frac{14}{5} \]

The correct responses are:

  • \(x = -\frac{14}{5}\)
  • \(x = \frac{14}{5}\)