To solve the equation \( 2x - 3x^2 + 3 = x^2 + 3 \), let’s first simplify it.
We can move all terms to one side of the equation to set it equal to zero:
\[ 2x - 3x^2 + 3 - x^2 - 3 = 0 \]
Combining like terms gives us:
\[ -4x^2 + 2x = 0 \]
We can factor out \(2x\):
\[ 2x(-2x + 1) = 0 \]
This gives us two potential solutions:
- \(2x = 0\) → \(x = 0\)
- \(-2x + 1 = 0\) → \(-2x = -1\) → \(x = \frac{1}{2}\)
Thus, the solutions to the equation are \(x = 0\) and \(x = \frac{1}{2}\).
However, the provided responses do not include 0 or \(\frac{1}{2}\). It seems there may have been an initial misinterpretation of the equation or the required values are not calculated correctly in accordance with the provided answers.
If we look analytically at the responses:
- \(-1\)
- \(\frac{3}{2}\)
- \(1\)
- \(-\frac{3}{2}\)
These solutions do not match the equivalent forms found. Please verify if there was a typo in the equation or if there are specific values we should interpret differently.
Upon careful examination, it appears the actual equation might have been misinterpreted. Please check the equation again and clarify, or proceed with only the observed solutions \(0\) and \(\frac{1}{2}\) from above.
If further assistance is required with accurate original input, I am here to help!