Let's solve each of the equations step by step.

Part (a) $4z - z(3 + i) = -1 + 3i$

1. Distribute $z$ in the equation: $4z - z(3 + i) = -1 + 3i$ $4z - 3z - iz = -1 + 3i$ $z(4 - 3 - i) = -1 + 3i$ $z(1 - i) = -1 + 3i$

2. Solve for $z$ by dividing both sides by $1 - i$: $z = \frac{-1 + 3i}{1 - i}$

3. Multiply the numerator and denominator by the conjugate of the denominator to simplify: $z = \frac{(-1 + 3i)(1 + i)}{(1 - i)(1 + i)}$ $z = \frac{-1(1 + i) + 3i(1 + i)}{1^2 - i^2}$ $z = \frac{-1 - i + 3i + 3i^2}{1 - (-1)}$ $z = \frac{-1 + 2i - 3}{2}$ $z = \frac{-4 + 2i}{2}$ $z = -2 + i$ So, $z = -2 + i$.

Part (b) $z(5 - 3i)^2 = 4 + 3i$

1. Expand the square on the left-hand side: $(5 - 3i)^2 = (5 - 3i)(5 - 3i)$ $= 25 - 15i - 15i + 9i^2$ $= 25 - 30i - 9$ $= 16 - 30i$

   So, the equation becomes: $z(16 - 30i) = 4 + 3i$

2. Solve for $z$ by dividing both sides by $16 - 30i$: $z = \frac{4 + 3i}{16 - 30i}$

3. Multiply the numerator and denominator by the conjugate of the denominator to simplify: $z = \frac{(4 + 3i)(16 + 30i)}{(16 - 30i)(16 + 30i)}$ $z = \frac{64 + 120i + 48i + 90i^2}{16^2 - (30i)^2}$ $z = \frac{64 + 168i - 90}{256 + 900}$ $z = \frac{-26 + 168i}{1156}$ $z = -\frac{13}{578} + \frac{84}{578}i$

   So, $z = -\frac{13}{578} + \frac{84}{578}i$.

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Summary of Solutions

• Part (a): $z = -2 + i$
• Part (b): $z = -\frac{13}{578} + \frac{84}{578}i$

Would you like any more details or have any questions?

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Further Questions:

1. How do you multiply complex numbers?
2. What is the conjugate of a complex number?
3. How do you rationalize a complex fraction?
4. How do you expand binomials involving complex numbers?
5. What are the properties of complex conjugates?
6. How can you add and subtract complex numbers?
7. What is the modulus of a complex number?
8. How do you represent complex numbers graphically?

Tip:

When dividing complex numbers, multiplying by the conjugate is a powerful technique to simplify the expression.