Gwen was asked to solve 3x^2-7=20 by factoring using the difference in square patterns. What are the correct solutions?

Answer 1

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:

\[ x = 3 \quad \text{and} \quad x = -3 \]