Student's completed worksheet demonstrating how to calculate the discriminant ($b^2 - 4ac$) for various quadratic equations.

Problem Analysis:

The task involves finding the discriminant of each quadratic equation and using it to determine the nature of the solutions (real or imaginary). The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by:

\[
\Delta = b^2 - 4ac
\]

- If \( \Delta > 0 \), the equation has two distinct real solutions.
- If \( \Delta = 0 \), the equation has one real solution (a repeated root).
- If \( \Delta < 0 \), the equation has two complex (imaginary) solutions.

Step-by-Step Solution:

#### Part 1: Find the value of the discriminant for each quadratic equation.

1. Equation 1: \( 6p^2 - 2p - 3 = 0 \)
- Here, \( a = 6 \), \( b = -2 \), \( c = -3 \).
- Discriminant:
\[
\Delta = b^2 - 4ac = (-2)^2 - 4(6)(-3) = 4 + 72 = 76
\]
- Discriminant: \( 76 \)

2. Equation 2: \( -2x^2 - x - 1 = 0 \)
- Here, \( a = -2 \), \( b = -1 \), \( c = -1 \).
- Discriminant:
\[
\Delta = b^2 - 4ac = (-1)^2 - 4(-2)(-1) = 1 - 8 = -7
\]
- Discriminant: \( -7 \)

3. Equation 3: \( -4m^2 - 4m + 5 = 0 \)
- Here, \( a = -4 \), \( b = -4 \), \( c = 5 \).
- Discriminant:
\[
\Delta = b^2 - 4ac = (-4)^2 - 4(-4)(5) = 16 + 80 = 96
\]
- Discriminant: \( 96 \)

4. Equation 4: \( 5b^2 + b - 2 = 0 \)
- Here, \( a = 5 \), \( b = 1 \), \( c = -2 \).
- Discriminant:
\[
\Delta = b^2 - 4ac = (1)^2 - 4(5)(-2) = 1 + 40 = 41
\]
- Discriminant: \( 41 \)

5. Equation 5: \( r^2 + 5r + 2 = 0 \)
- Here, \( a = 1 \), \( b = 5 \), \( c = 2 \).
- Discriminant:
\[
\Delta = b^2 - 4ac = (5)^2 - 4(1)(2) = 25 - 8 = 17
\]
- Discriminant: \( 17 \)

6. Equation 6: \( 2p^2 + 5p - 4 = 0 \)
- Here, \( a = 2 \), \( b = 5 \), \( c = -4 \).
- Discriminant:
\[
\Delta = b^2 - 4ac = (5)^2 - 4(2)(-4) = 25 + 32 = 57
\]
- Discriminant: \( 57 \)

#### Part 2: Find the discriminant of each quadratic equation and state the number of real and imaginary solutions.

7. Equation 7: \( 9n^2 - 3n - 8 = -10 \)
- First, rewrite the equation in standard form:
\[
9n^2 - 3n - 8 + 10 = 0 \implies 9n^2 - 3n + 2 = 0
\]
- Here, \( a = 9 \), \( b = -3 \), \( c = 2 \).
- Discriminant:
\[
\Delta = b^2 - 4ac = (-3)^2 - 4(9)(2) = 9 - 72 = -63
\]
- Discriminant: \( -63 \)
- Since \( \Delta < 0 \), there are two imaginary solutions.

8. Equation 8: \( -2x^2 - 8x - 14 = -6 \)
- First, rewrite the equation in standard form:
\[
-2x^2 - 8x - 14 + 6 = 0 \implies -2x^2 - 8x - 8 = 0
\]
- Here, \( a = -2 \), \( b = -8 \), \( c = -8 \).
- Discriminant:
\[
\Delta = b^2 - 4ac = (-8)^2 - 4(-2)(-8) = 64 - 64 = 0
\]
- Discriminant: \( 0 \)
- Since \( \Delta = 0 \), there is one real solution (a repeated root).

Final Answers:

\[
\boxed{
\begin{array}{ll}
1) & 76 \\
2) & -7 \\
3) & 96 \\
4) & 41 \\
5) & 17 \\
6) & 57 \\
7) & -63 \quad (\text{2 imaginary solutions}) \\
8) & 0 \quad (\text{1 real solution})
\end{array}
}
\]