Let's work through these problems one by one:

Problem 9

The equation given is: $8 - 6m^2 = -88$

1. Move the 8 to the other side: $-6m^2 = -88 - 8$ $-6m^2 = -96$

2. Divide by -6: $m^2 = \frac{96}{6} = 16$

3. Take the square root: $m = \pm \sqrt{16} = \pm 4$

Solution: $m = \pm 4$.

Problem 10

The equation given is: $3p^2 - 4 = -6$

1. Move -4 to the other side: $3p^2 = -6 + 4$ $3p^2 = -2$

Since this results in a negative on the right, and $p^2$ cannot equal a negative number in the real number system, there is no real solution for $p$ in this equation.

Simplification Problems

Let's simplify each of the complex number expressions.

Problem 11

$(-4 - 7i) - (4 + 5i) - (2 - i)$

1. Distribute the minus signs: $= -4 - 7i - 4 - 5i - 2 + i$

2. Combine like terms:

   • Real part: $-4 - 4 - 2 = -10$
   • Imaginary part: $-7i - 5i + i = -11i$

Solution: $-10 - 11i$.

Problem 12

$(-1 + i) - (-7 + 4i) - 5$

1. Distribute the minus signs: $= -1 + i + 7 - 4i - 5$

2. Combine like terms:

   • Real part: $-1 + 7 - 5 = 1$
   • Imaginary part: $i - 4i = -3i$

Solution: $1 - 3i$.

Problem 13

$(-1 + 6i) + (5 - 2i) - (8i)$

1. Distribute any signs: $= -1 + 6i + 5 - 2i - 8i$

2. Combine like terms:

   • Real part: $-1 + 5 = 4$
   • Imaginary part: $6i - 2i - 8i = -4i$

Solution: $4 - 4i$.

Problem 14

$(1 + 6i) + (6 - 2i) - (-7 + 5i)$

1. Distribute the minus sign: $= 1 + 6i + 6 - 2i + 7 - 5i$

2. Combine like terms:

   • Real part: $1 + 6 + 7 = 14$
   • Imaginary part: $6i - 2i - 5i = -i$

Solution: $14 - i$.

Would you like further details on any specific problem or concept?

Related Questions

1. How do you handle equations with no real solutions?
2. What is the method for simplifying expressions with multiple complex numbers?
3. Can imaginary solutions arise in quadratic equations?
4. What is the significance of combining like terms in complex numbers?
5. How can complex numbers be represented graphically?

Tip

When simplifying complex numbers, always combine real parts and imaginary parts separately for clarity.