Square roots and the solvability of quadratic equations in ℂ Let c=a+i b ≠0 be a given complex n

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Square roots and the solvability of quadratic equations in ℂ...

Square roots and the solvability of quadratic equations in $\mathbb{C}$ Let $c=a+\mathrm{i} b \neq 0$ be a given complex number. By splitting it into its real and imaginary part show that there are exactly two complex numbers $z_{1}$ and $z_{2}$ such that $$ z_{1}^{2}=z_{2}^{2}=c . \text { One has } z_{2}=-z_{1} $$ $\left(z_{1}\right.$ and $z_{2}$ are called the square roots of $c$ ) For example, determine the square roots of $$ 5+7 \mathrm{i}, \quad \text { and } \quad \sqrt{2}+\mathrm{i} \sqrt{2} $$ Use also polar coordinates for this exercise. Furthermore, show that a quadratic equation $$ z^{2}+\alpha z+\beta=0, \quad \alpha, \beta \in \mathbb{C} \text { arbitrary } $$ always has at most two solutions $z_{1}, z_{2} \in \mathrm{C}$.

Square roots and the solvability of quadratic equations in $\mathbb{C}$
Let $c=a+\mathrm{i} b \neq 0$ be a given complex number. By splitting it into its real and imaginary part show that there are exactly two complex numbers $z_{1}$ and $z_{2}$ such that
$$
z_{1}^{2}=z_{2}^{2}=c . \text { One has } z_{2}=-z_{1}
$$
$\left(z_{1}\right.$ and $z_{2}$ are called the square roots of $c$ ) For example, determine the square roots of
$$
5+7 \mathrm{i}, \quad \text { and } \quad \sqrt{2}+\mathrm{i} \sqrt{2}
$$
Use also polar coordinates for this exercise. Furthermore, show that a quadratic equation
$$
z^{2}+\alpha z+\beta=0, \quad \alpha, \beta \in \mathbb{C} \text { arbitrary }
$$
always has at most two solutions $z_{1}, z_{2} \in \mathrm{C}$.

Key Concepts

Complex Number Representations

Complex numbers can be expressed in more than one form, most notably the rectangular form (a + i b) and the polar form (r e^(i?)). Each representation has advantages: the rectangular form is useful for addition and subtraction while the polar form simplifies multiplication, division, and extraction of roots. Understanding these representations is fundamental for performing algebraic manipulations and solving equations in the complex plane.

Square Roots in the Complex Plane

Finding the square roots of a complex number involves determining the complex numbers that, when squared, yield the original number. A non-zero complex number has two distinct square roots which are negatives of each other. This concept requires splitting the complex number into its real and imaginary components, or alternatively, expressing it in polar form to take advantage of de Moivre's theorem.

Polar Form and Euler's Formula

The polar coordinate representation of complex numbers, expressed as r e^(i?), leverages Euler's formula, which relates complex exponentials to trigonometric functions. This form greatly simplifies the process of extracting roots and solving equations by converting multiplicative and exponential operations into more manageable angular operations.

Quadratic Equations in the Complex Domain

Quadratic equations with complex coefficients extend the familiar quadratic formula to the complex field. The key insight is that, just as in the real number system, a quadratic equation over the complex numbers can have at most two solutions. This fact is underlined by the fundamental theorem of algebra, which guarantees that every non-constant polynomial equation has as many roots as its degree, when counted with multiplicity.

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