Question
Gwen was asked to solve 3x2−20=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=3
x equals 3
x=9, x=−9
x equals 9 , x equals negative 9
x=−3
Responses
x=−3, x=3
x equals negative 3 , x equals 3
x=3
x equals 3
x=9, x=−9
x equals 9 , x equals negative 9
x=−3
Answers
GPT-4o mini
To solve the equation \(3x^2 - 20 = 7\) by factoring, we first need to rearrange the equation:
1. Move all terms to one side:
\[
3x^2 - 20 - 7 = 0 \quad \Rightarrow \quad 3x^2 - 27 = 0
\]
2. Factor out the common factor of 3:
\[
3(x^2 - 9) = 0
\]
3. Next, we can use the difference of squares pattern, which states that \(a^2 - b^2 = (a - b)(a + b)\). In this case, \(x^2 - 9\) can be factored as follows:
\[
x^2 - 9 = (x - 3)(x + 3)
\]
4. Putting it all together, we have:
\[
3(x - 3)(x + 3) = 0
\]
5. Setting each factor equal to zero gives us the solutions:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
Thus, the correct solutions to the equation are:
**x = -3, x = 3**.
1. Move all terms to one side:
\[
3x^2 - 20 - 7 = 0 \quad \Rightarrow \quad 3x^2 - 27 = 0
\]
2. Factor out the common factor of 3:
\[
3(x^2 - 9) = 0
\]
3. Next, we can use the difference of squares pattern, which states that \(a^2 - b^2 = (a - b)(a + b)\). In this case, \(x^2 - 9\) can be factored as follows:
\[
x^2 - 9 = (x - 3)(x + 3)
\]
4. Putting it all together, we have:
\[
3(x - 3)(x + 3) = 0
\]
5. Setting each factor equal to zero gives us the solutions:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
Thus, the correct solutions to the equation are:
**x = -3, x = 3**.