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The quadratic formula is a powerful tool for solving quadratic equations. A quadratic equation is generally written as \( ax^2 + bx + c = 0 \).
The quadratic formula helps you find the solutions to this equation, referred to as the roots.
It is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
The expression inside the square root, \( b^2 - 4ac \), is called the discriminant, and it determines the nature of the roots.
The plus-minus symbol (\pm) indicates that there are usually two solutions: one for the plus and one for the minus.

Complex numbers extend the familiar real number system.
They are written in the form \( z = x + yi \), where \( x \) and \( y \) are real numbers, and \( i \) is the imaginary unit, which satisfies \( i^2 = -1 \).
This means complex numbers have both a real part (x) and an imaginary part (y).
For example, in the quadratic equation \( z^2 + iz - (2 + 6i) = 0 \), the solutions are found in the set of complex numbers.

The discriminant is an essential part of the quadratic formula.
It is the term under the square root: \( b^2 - 4ac \).
The value of the discriminant tells us about the nature of the roots:

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is one real root (a repeating root).
• If the discriminant is negative, the roots are complex conjugates, meaning they are not real numbers.

In our equation, the discriminant \( i^2 - 4 \cdot 1 \cdot (-2 - 6i) = 7 + 24i \) is a complex number, leading to complex roots.

To simplify the square root of a complex number like
\( \sqrt{7 + 24i} \), we use polar form.
A complex number can be represented as
\( re^{i\theta} \), where \( r \) is the modulus and \( \theta \) is the argument.
The modulus \( r \) is the distance from the origin in the complex plane and is calculated as \[ r = \sqrt{x^2 + y^2} \].
In our example, \( r = \sqrt{7^2 + 24^2} = 25 \).
The argument \( \theta \) is the angle with the positive real axis and is found using \[ \theta = \tan^{-1} \left( \frac{y}{x} \right) \].
Once we have \( r \) and \( \theta \), we can find the square root:
\( \sqrt{re^{i\theta}} = \sqrt{r} \cdot e^{i \theta / 2} \). In our equation, this becomes \( \sqrt{7 + 24i} \approx 3 + 4i \).
This makes it easier to continue solving the quadratic equation.