Let's analyze each quadratic equation, find its discriminant, and then determine the number of real or imaginary solutions based on that discriminant.

The discriminant $\Delta$ of a quadratic equation of the form $ax^2 + bx + c = 0$ is given by:

$\Delta = b^2 - 4ac$

Based on the value of $\Delta$:

• If $\Delta > 0$, the equation has 2 distinct real solutions.
• If $\Delta = 0$, the equation has 1 real solution (a repeated root).
• If $\Delta < 0$, the equation has 2 complex (imaginary) solutions.

Now, let's calculate the discriminants for each of the quadratic equations:

1. $-2x^2 - 8x - 14 = 6$

Rewrite the equation as: $-2x^2 - 8x - 14 - 6 = 0$ $-2x^2 - 8x - 20 = 0$ Here, $a = -2$, $b = -8$, and $c = -20$.

Discriminant: $\Delta = (-8)^2 - 4(-2)(-20) = 64 - 160 = -96$ Since $\Delta < 0$, this equation has 2 complex (imaginary) solutions.

2. $9n^2 - 3n - 8 = -10$

Rewrite the equation as: $9n^2 - 3n - 8 + 10 = 0$ $9n^2 - 3n + 2 = 0$ Here, $a = 9$, $b = -3$, and $c = 2$.

Discriminant: $\Delta = (-3)^2 - 4(9)(2) = 9 - 72 = -63$ Since $\Delta < 0$, this equation has 2 complex (imaginary) solutions.

3. $9m^2 + 6m + 6 \geq 5$

Rewrite the inequality as: $9m^2 + 6m + 6 - 5 \geq 0$ $9m^2 + 6m + 1 \geq 0$ Here, $a = 9$, $b = 6$, and $c = 1$.

Discriminant: $\Delta = 6^2 - 4(9)(1) = 36 - 36 = 0$ Since $\Delta = 0$, this quadratic inequality has 1 real solution (a repeated root). However, because it's an inequality, the quadratic is non-negative (always greater or equal to 0) for all values of $m$, meaning all real numbers will satisfy the inequality.

4. $9b^2 = -8b + 8$

Rewrite the equation as: $9b^2 + 8b - 8 = 0$ Here, $a = 9$, $b = 8$, and $c = -8$.

Discriminant: $\Delta = 8^2 - 4(9)(-8) = 64 + 288 = 352$ Since $\Delta > 0$, this equation has 2 distinct real solutions.

Summary of Results:

1. Equation 1: 2 complex (imaginary) solutions.
2. Equation 2: 2 complex (imaginary) solutions.
3. Equation 3: 1 real solution (non-negative for all $m$).
4. Equation 4: 2 distinct real solutions.

Would you like further details or clarifications on any of these steps?

Here are 5 related questions:

1. How do you solve quadratic equations using the quadratic formula?
2. What is the significance of the discriminant in determining solution types?
3. Can a quadratic inequality have complex solutions?
4. How do you solve systems involving quadratic equations?
5. What are some real-world applications of quadratic equations?

Tip: When solving a quadratic equation, always simplify the equation first to standard form $ax^2 + bx + c = 0$ to apply formulas effectively.