Compute and plot the roots of the following quadratic equations: a. \(z^{2}+2 z+2=0\) b. \(z^{2}-2 z+2=0\) c. \(z^{2}+2=0\) For each equation, check that \(2 \operatorname{Re}\left[z_{1,2}\right]=-\frac{b}{a}\) and \(\left|z_{1,2}\right|^{2}=\frac{c}{a}\).

A quadratic equation is generally given in the form ax^2 + bx + c = 0. Here, a, b, and c are called the coefficients. To find the roots or solutions of these equations, one can use various methods, like factoring, completing the square, or the widely-used quadratic formula.

Roots are essentially the values of x for which the quadratic equation holds true. They are found by solving the equation, and can be both real and complex numbers. For instance, in our exercises:

- For the equation z^2 + 2z + 2 = 0, the roots are -1 ± i (complex numbers).
- For z^2 - 2z + 2 = 0, the roots are 1 ± i.
- And z^2 + 2 = 0 has roots ±i.

In these examples, the roots are complex because the parts under the square root become negative, leading us to a result involving the imaginary unit, i, (where i is defined as \(\sqrt{-1}\)).

The complex plane is a useful way to visualize complex numbers, which have both real and imaginary parts. It consists of two axes:

- The horizontal axis (x-axis) is the Real Axis.
- The vertical axis (y-axis) is the Imaginary Axis.

Each complex number can be represented as a point on this plane. For example:

- The complex number -1 + i is plotted at (-1, 1).
- The complex number 1 - i is plotted at (1, -1).
- Similarly, the number i is plotted at (0, 1), and -i at (0, -1).

By plotting these roots, we gain intuitive insights into their properties and relationships. In the solutions above, plotting:

- -1 ± i gives us points (-1, 1) and (-1, -1).
- 1 ± i gives us points (1, 1) and (1, -1).
- ±i are at (0, 1) and (0, -1).

The quadratic formula is a powerful tool for finding roots of quadratic equations. It is given by:

\[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, the term under the square root is called the discriminant. This formula works regardless of whether the roots are real or complex.

Breaking down the steps of our solutions:

- For z^2 + 2z + 2 = 0, we applied the formula to find the roots:
\[ z = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]
leading to \[ z = -1 \pm i \].
- For z^2 - 2z + 2 = 0, we solved:
\[ z = \frac{2 \pm \sqrt{(-2)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]
resulting in \[ z = 1 \pm i \].
- For z^2 + 2 = 0, the root calculation was:
\[ z = \frac{0 \pm \sqrt{0^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]
yielding \[ z = \pm i \].

In quadratic equations, the coefficients a, b, and c play crucial roles:

• a determines the direction and width of the parabola formed by the equation. If a is positive, the parabola opens upwards, and if negative, it opens downwards.


• b affects the position of the parabola along the x-axis.


• c is the y-intercept, the point where the parabola intersects the y-axis.

Understanding the coefficients:

- For z^2 + 2z + 2 = 0, we have a = 1, b = 2, and c = 2.
- For z^2 - 2z + 2 = 0, here a = 1, b = -2, and c = 2.
- For z^2 + 2 = 0, a = 1, b = 0, and c = 2.

These coefficients help us to understand not just the parabola’s shape but also its roots and properties, like the sum and product of the roots. Knowledge of coefficients allows verifying conditions such as '2 Re[z] = -b/a' and |z|^2 = c/a as we did in step 4 of the solutions.