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To determine the number of real solutions for each quadratic equation, we examine the discriminant of each equation. The discriminant of a quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex] is given by the formula [tex]\( \Delta = b^2 - 4ac \)[/tex].
Depending on the value of the discriminant:
- If [tex]\( \Delta > 0 \)[/tex], there are two distinct real solutions.
- If [tex]\( \Delta = 0 \)[/tex], there is exactly one real solution.
- If [tex]\( \Delta < 0 \)[/tex], there are no real solutions (instead, there are two complex solutions).
Given the quadratic equations:
1. [tex]\( y = 3x^2 - 6x + 3 \)[/tex]
2. [tex]\( y = -2x^2 + 9x - 11 \)[/tex]
3. [tex]\( y = -x^2 - 4x + 7 \)[/tex]
Let's analyze these equations one by one.
### For the equation [tex]\( y = 3x^2 - 6x + 3 \)[/tex]:
The coefficients are [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 3 \)[/tex].
- The discriminant is [tex]\( \Delta = (-6)^2 - 4 \cdot 3 \cdot 3 = 36 - 36 = 0 \)[/tex].
- Since [tex]\(\Delta = 0 \)[/tex], this equation has one real solution.
### For the equation [tex]\( y = -2x^2 + 9x - 11 \)[/tex]:
The coefficients are [tex]\( a = -2 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = -11 \)[/tex].
- The discriminant is [tex]\( \Delta = 9^2 - 4 \cdot (-2) \cdot (-11) = 81 - 88 = -7 \)[/tex].
- Since [tex]\(\Delta < 0 \)[/tex], this equation has no real solutions but two complex solutions.
### For the equation [tex]\( y = -x^2 - 4x + 7 \)[/tex]:
The coefficients are [tex]\( a = -1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = 7 \)[/tex].
- The discriminant is [tex]\( \Delta = (-4)^2 - 4 \cdot (-1) \cdot 7 = 16 + 28 = 44 \)[/tex].
- Since [tex]\(\Delta > 0 \)[/tex], this equation has two distinct real solutions.
Based on these calculations, we can match each equation with its number of unique solutions as follows:
- [tex]\( y = 3x^2 - 6x + 3 \)[/tex]: One Real Solution
- [tex]\( y = -2x^2 + 9x - 11 \)[/tex]: No Real Solutions (Two Complex Solutions)
- [tex]\( y = -x^2 - 4x + 7 \)[/tex]: Two Real Solutions
Thus, the correct filled table is:
\begin{tabular}{|l|l|}
\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] \\
\hline
\end{tabular}
Depending on the value of the discriminant:
- If [tex]\( \Delta > 0 \)[/tex], there are two distinct real solutions.
- If [tex]\( \Delta = 0 \)[/tex], there is exactly one real solution.
- If [tex]\( \Delta < 0 \)[/tex], there are no real solutions (instead, there are two complex solutions).
Given the quadratic equations:
1. [tex]\( y = 3x^2 - 6x + 3 \)[/tex]
2. [tex]\( y = -2x^2 + 9x - 11 \)[/tex]
3. [tex]\( y = -x^2 - 4x + 7 \)[/tex]
Let's analyze these equations one by one.
### For the equation [tex]\( y = 3x^2 - 6x + 3 \)[/tex]:
The coefficients are [tex]\( a = 3 \)[/tex], [tex]\( b = -6 \)[/tex], and [tex]\( c = 3 \)[/tex].
- The discriminant is [tex]\( \Delta = (-6)^2 - 4 \cdot 3 \cdot 3 = 36 - 36 = 0 \)[/tex].
- Since [tex]\(\Delta = 0 \)[/tex], this equation has one real solution.
### For the equation [tex]\( y = -2x^2 + 9x - 11 \)[/tex]:
The coefficients are [tex]\( a = -2 \)[/tex], [tex]\( b = 9 \)[/tex], and [tex]\( c = -11 \)[/tex].
- The discriminant is [tex]\( \Delta = 9^2 - 4 \cdot (-2) \cdot (-11) = 81 - 88 = -7 \)[/tex].
- Since [tex]\(\Delta < 0 \)[/tex], this equation has no real solutions but two complex solutions.
### For the equation [tex]\( y = -x^2 - 4x + 7 \)[/tex]:
The coefficients are [tex]\( a = -1 \)[/tex], [tex]\( b = -4 \)[/tex], and [tex]\( c = 7 \)[/tex].
- The discriminant is [tex]\( \Delta = (-4)^2 - 4 \cdot (-1) \cdot 7 = 16 + 28 = 44 \)[/tex].
- Since [tex]\(\Delta > 0 \)[/tex], this equation has two distinct real solutions.
Based on these calculations, we can match each equation with its number of unique solutions as follows:
- [tex]\( y = 3x^2 - 6x + 3 \)[/tex]: One Real Solution
- [tex]\( y = -2x^2 + 9x - 11 \)[/tex]: No Real Solutions (Two Complex Solutions)
- [tex]\( y = -x^2 - 4x + 7 \)[/tex]: Two Real Solutions
Thus, the correct filled table is:
\begin{tabular}{|l|l|}
\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] \\
\hline
\end{tabular}
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