Gwen was asked to solve 3x^2−20=7 by factoring using the difference of squares pattern. What is/are the correct solution(s)?(1 point)

Question

Responses

x=3
x equals 3

x=−3
x equals negative 3

x=9, x=−9
x equals 9 , x equals negative 9

x=−3, x=3
x equals negative 3 , x equals 3

Answer 1

To solve the equation \( 3x^2 - 20 = 7 \) by factoring, we first rearrange it:

1. Subtract 7 from both sides: \[ 3x^2 - 20 - 7 = 0 \] Simplifying gives: \[ 3x^2 - 27 = 0 \]

2. Factor out the common factor (which is 3): \[ 3(x^2 - 9) = 0 \]

3. Now, recognize that \( x^2 - 9 \) is a difference of squares: \[ x^2 - 9 = (x - 3)(x + 3) \]

4. Thus, our equation can be factored as: \[ 3(x - 3)(x + 3) = 0 \]

5. Setting each factor equal to zero gives us the solutions: \[ x - 3 = 0 \quad \Rightarrow \quad x = 3 \] \[ x + 3 = 0 \quad \Rightarrow \quad x = -3 \]

So the correct solutions are \( x = 3 \) and \( x = -3 \).

From the provided responses, the correct answer is:

x = −3, x = 3 (x equals negative 3, x equals 3).