For any complex number \[ z = x + iy ,\]
\[ z \bar{z} = (x + iy)(x -iy) \]
\[ = x^{2} + ixy - ixy + y^{2} = x^{2} + y^{2} .\]
That is, any complex number times its complex conjugate equals the square of its real part plus the square of its imaginary part.
Because both the real and imaginary parts of any complex number are real:
\[ z \bar{z} = x^{2} + y^{2} \geq 0, \forall z \in \mathbb{C}.\]
We define the positive square root of \[ z \bar{z}: \]
\[ +\sqrt{z\bar{z}} : = |z| ,\]
to be the Modulus of the complex number \[ z .\]
Two complex numbers of the form:
\[ z_+ = x + iy ,\]
and,
\[ z_- = x - iy ,\]
are called Complex Conjugates.
Recall that a polynomial of degree two is called a Quadratic Polynomial.
A general quadratic polynomial (over the real numbers) is of the form:
\[ az^{2} + bz + c = 0. \]
The formula for solving for the roots of this equation is:
\[ z = \frac{-b \pm \sqrt{b^{2} -4ac}}{2a} .\]
Take a pencil and place it along the edge of a ruler, so that the ruler and the pencil are parallel, and so that the end of the eraser is at zero, and the point of the pencil points towards increasing numbers on the ruler.
The official formula is: D = R * T
Distance = Rate x Time
This can also be flipped around to be expressed as R = D / T (Rate = Distance / Time)
Or it can be flipped around yet again to be expressed as T = D / R (Time = Distance / Rate)
This formula is completely liquid and can be "remelded" as needed depending on what you're working on.
4x - 2x - 5 = 4 + 6x + 3 |
Goal is to isolate x on one side to be able to solve. |
2x - 5 = 7 + 6x |
Combine like terms. |
2x - 5 + 5 = 7 + 6x + 5 |
Add 5 to each side. |
2x = 12 + 6x |
Subtract 6x from each side. |
-4x = 12 |
Then, divide both sides by -4 |
x = -3 |
CHECK your solution by plugging back into the original equation. |
Disclaimer: I did not create nor do I own these videos. I have simply embedded them, courtesy of YouTube. (But I do think this teacher does a fantastic job with his video tutorial series.)