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How do you solve this question without a calculator?

$z^2 = 4- 3i$. Find $z$.

I know how to find the answer to this question using de Moivre's theorem with a calculator. What I do is I start out by finding the angle of $z^2$ by finding $\tan^{-1}(-3/4)$ as a decimal and I then I solve the question using the decimal with my calculator.

I need to be able to do this question without a calculator if possible.

Supriyo
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user110069
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2 Answers2

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HINT

Letting $z=a+bi \ (a,b\in\mathbb R)$, you'll get $$(a+bi)^2=4-3i\iff (a^2-b^2)+2abi=4-3i.$$ Hence, comparing both real part and imaginary part will give you $$a^2-b^2=4,$$ $$2ab=-3.$$

Two equations with two variables, so you can find $(a,b)$.

mathlove
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  • If you solve for $b$ in the second and substitute in the first equation, you will get a quadratic in $a^2$. Once you get $a^2$, you can pick $\pm$ in the square root to get the two square roots of $z$ – user44197 Jan 04 '14 at 06:20
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Write $z = a + bi$, where $a$ and $b$ are real numbers, and expand $z^2$. Equate the real and imaginary part of the resulting expression with $4$ and $-3$, respectively. This gives you a system to solve for the real numbers $a$ and $b$. There will be two possibilities.

user119003
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