To determine the number of unique solutions for each quadratic equation, we use the discriminant [tex]\( \Delta \)[/tex] given by the equation [tex]\(\Delta = b^2 - 4ac \)[/tex]. Here is the result for each equation:
1. For the equation [tex]\( y = 3x^2 - 6x + 3 \)[/tex]:
- The discriminant is zero, which means there is one real solution.
- So, this equation goes under "One Real Solution".
2. For the equation [tex]\( y = -x^2 - 4x + 7 \)[/tex]:
- The discriminant is positive, which means there are two real solutions.
- So, this equation goes under "Two Real Solutions".
3. For the equation [tex]\( y = -2x^2 + 9x - 11 \)[/tex]:
- The discriminant is negative, which means there are two complex solutions.
- So, this equation goes under "Two Complex Solutions".
Thus, the table will be filled as follows:
\begin{tabular}{|l|c|}
\hline Two Real Solutions & One Real Solution \\
\hline [tex]\( y = -x^2 - 4x + 7 \)[/tex] & [tex]\( y = 3x^2 - 6x + 3 \)[/tex] \\
& \\
\hline One Complex Solution & Two Complex Solutions \\
\hline & [tex]\( y = -2x^2 + 9x - 11 \)[/tex] \\
& \\
& \\
& \\
& \\
& \\
& \\
& \\
&
\end{tabular}
