We have a list of math problems to solve. We will solve problem 24, 25, 26, 29 and 30. Problem 24: Simplify the expression $(1-2i) + (-2-i)$. Problem 25: Simplify the expression $\frac{-2+i}{-4-5i}$. Problem 26: Find the solutions of the equation $\frac{1}{2}x^2 - 2x + 3 = 0$. Problem 29: Solve for $x$ in the equation $108x^2 = 147$. Problem 30: Solve for $x$ in the equation $49x^2 + 36 = 0$.

1. Problem Description

We have a list of math problems to solve. We will solve problem 24, 25, 26, 29 and

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0. Problem 24: Simplify the expression $(1-2i) + (-2-i)$.

Problem 25: Simplify the expression $\frac{-2+i}{-4-5i}$.

Problem 26: Find the solutions of the equation $\frac{1}{2}x^2 - 2x + 3 = 0$.

Problem 29: Solve for $x$ in the equation $108x^2 = 147$.

Problem 30: Solve for $x$ in the equation $49x^2 + 36 = 0$.

2. Solution Steps

Problem 24:

Combine the real and imaginary parts.

$(1-2i) + (-2-i) = (1-2) + (-2i - i) = -1 - 3i$

Problem 25:

Multiply the numerator and denominator by the conjugate of the denominator.

$\frac{-2+i}{-4-5i} = \frac{(-2+i)(-4+5i)}{(-4-5i)(-4+5i)}$

$= \frac{8 - 10i - 4i + 5i^2}{16 - 25i^2} = \frac{8 - 14i - 5}{16 + 25} = \frac{3-14i}{41} = \frac{3}{41} - \frac{14}{41}i$

Problem 26:

We have the quadratic equation $\frac{1}{2}x^2 - 2x + 3 = 0$. Multiply by 2 to get $x^2 - 4x + 6 = 0$.

Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, with $a=1$, $b=-4$, and $c=6$.

$x = \frac{4 \pm \sqrt{(-4)^2 - 4(1)(6)}}{2(1)} = \frac{4 \pm \sqrt{16 - 24}}{2} = \frac{4 \pm \sqrt{-8}}{2} = \frac{4 \pm 2i\sqrt{2}}{2} = 2 \pm i\sqrt{2}$

Problem 29:

Solve for $x$ in the equation $108x^2 = 147$.

$x^2 = \frac{147}{108} = \frac{49}{36}$

$x = \pm \sqrt{\frac{49}{36}} = \pm \frac{7}{6}$

Problem 30:

Solve for $x$ in the equation $49x^2 + 36 = 0$.

$49x^2 = -36$

$x^2 = -\frac{36}{49}$

$x = \pm \sqrt{-\frac{36}{49}} = \pm \frac{6}{7}i$

3. Final Answer

Problem 24:

$-1 - 3i$

Problem 25:

$\frac{3}{41} - \frac{14}{41}i$

Problem 26:

$2 \pm i\sqrt{2}$

Problem 29:

$\pm \frac{7}{6}$

Problem 30:

$\pm \frac{6}{7}i$