Question
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) Responses x=−3 , x=3 x equals negative 3 , x equals 3 x=9 , x=−9 x equals 9 , x equals negative 9 x=3 x equals 3 x=−3 x equals negative 3
Answers
GPT-4o mini
To solve the equation \(3x^2 - 20 = 7\) by factoring, we first need to set the equation to zero:
\[
3x^2 - 20 - 7 = 0
\]
This simplifies to:
\[
3x^2 - 27 = 0
\]
Next, we can factor out the common term:
\[
3(x^2 - 9) = 0
\]
Now we can recognize that \(x^2 - 9\) is a difference of squares, which can be factored further:
\[
3(x - 3)(x + 3) = 0
\]
Setting each factor equal to zero gives us the solutions:
1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x + 3 = 0 \Rightarrow x = -3\)
Thus, the solutions to the equation are:
\[
x = 3 \quad \text{and} \quad x = -3
\]
Among the provided responses, the correct answers are:
- \(x = 3, x = -3\)
- \(x \text{ equals } 3, x \text{ equals } -3\)
So either of those representations of the solution is correct.
\[
3x^2 - 20 - 7 = 0
\]
This simplifies to:
\[
3x^2 - 27 = 0
\]
Next, we can factor out the common term:
\[
3(x^2 - 9) = 0
\]
Now we can recognize that \(x^2 - 9\) is a difference of squares, which can be factored further:
\[
3(x - 3)(x + 3) = 0
\]
Setting each factor equal to zero gives us the solutions:
1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x + 3 = 0 \Rightarrow x = -3\)
Thus, the solutions to the equation are:
\[
x = 3 \quad \text{and} \quad x = -3
\]
Among the provided responses, the correct answers are:
- \(x = 3, x = -3\)
- \(x \text{ equals } 3, x \text{ equals } -3\)
So either of those representations of the solution is correct.