What is the degree of a quadratic function f? What is the standard form of a quadratic function? How do you put a quadratic function into standard form?

A little bit different quadratic Gauss sum
Could anyone give me a direction on how to demonstrate that
${\displaystyle \sum limit{s}_{k=0}^{N-1}{e}^{-i\frac{\pi }{N}{(k+C)}^2}=\sqrt{N}{e}^{-i\frac{\pi }{4}},}$
If ${\displaystyle N\in \mathbb{N}}$ is even, for any ${\displaystyle C\in \mathbb{Z}}$.
Unfortunately, the left side of the above equation cannot be exactly defined as a quadratic Gauss sum since it does not have the term 2 within the argument of e and the summation coefficient is shifted by C.

Please, help me to solve this equation. No advanced math should be needed.
${\displaystyle 3x^2-4x+\sqrt{3x^2-4x-6}=18}$

To solve: ${\displaystyle \frac{4t^2}{5}=\frac{t}{5}+\frac{3}{10}}$

Find all real values of a such that ${\displaystyle x^2+(a+i)x-5i=0}$ has at least one real solution.

Is an equation only a parabola if it is quadratic? Could the graph of ${\displaystyle y=x^{1.65}}$ be described as parabolic in shape?

Solve the following linear and quadratic systems of equations:
y + 3x = -2 or 3x +2 = y
y = ${\displaystyle x^2}$