Asked by LLL
Gwen was asked to solve 3x^2-7=20 by factoring using the difference in square patterns. What are the correct solutions?
Answers
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Answered by
GPT-4o mini
To solve the equation \(3x^2 - 7 = 20\) by factoring, we first need to rearrange the equation into standard form.
1. Subtract 20 from both sides of the equation:
\[
3x^2 - 7 - 20 = 0
\]
This simplifies to:
\[
3x^2 - 27 = 0
\]
2. Next, we can factor out the common factor (which is 3):
\[
3(x^2 - 9) = 0
\]
3. Now, we can set the equation inside the parentheses to zero:
\[
x^2 - 9 = 0
\]
4. Notice that this is a difference of squares, which can be factored further:
\[
(x - 3)(x + 3) = 0
\]
5. Now we can find the solutions by setting each factor equal to zero:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
Thus, the correct solutions to the equation \(3x^2 - 7 = 20\) are:
\[
x = 3 \quad \text{and} \quad x = -3
\]
1. Subtract 20 from both sides of the equation:
\[
3x^2 - 7 - 20 = 0
\]
This simplifies to:
\[
3x^2 - 27 = 0
\]
2. Next, we can factor out the common factor (which is 3):
\[
3(x^2 - 9) = 0
\]
3. Now, we can set the equation inside the parentheses to zero:
\[
x^2 - 9 = 0
\]
4. Notice that this is a difference of squares, which can be factored further:
\[
(x - 3)(x + 3) = 0
\]
5. Now we can find the solutions by setting each factor equal to zero:
\[
x - 3 = 0 \quad \Rightarrow \quad x = 3
\]
\[
x + 3 = 0 \quad \Rightarrow \quad x = -3
\]
Thus, the correct solutions to the equation \(3x^2 - 7 = 20\) are:
\[
x = 3 \quad \text{and} \quad x = -3
\]
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