Quadratic Function Real Life Examples and Applications

How to systematically find roots of ${\displaystyle x^2-x-132=0}$ with Po-Shen Loh's method?

${\displaystyle f\left(x\right)=a{x}^2+bx+c}$ are positive, and ${\displaystyle a+b+c=1}$
Prove ${\displaystyle f\left(x_1\right)f\left(x_2\right)\cdots f\left(x_n\right)\ge 1}$ is true for all positive x, satisying ${\displaystyle x_1{x}_2\cdots x_n=1}$

How to solve quadratic inequalities with two boundaries
What is possible values of x if the perimeter of some rectangle tile is between 20 cm and 54 cm.

How to solve a quadratic equation with Binary coefficients?
${\displaystyle 5x^2-50x+125=0}$
It has the roots ${\displaystyle x_1=5}$ and ${\displaystyle x_2=5}$. But now, convert these coefficients into binary:
${\displaystyle 101x^2-110010x+1111101=0}$
Solve this quadratic equation with binary coefficients.

How to solve a quadratic that needs complex numbers?
${\displaystyle x^2-2x\cos \left(\alpha \right)+1}$

${\displaystyle (\frac{r}{\sqrt{r^2-1}},\frac{r}{\sqrt{r^2+1}})}$ are roots of the equation ${\displaystyle x^2-bx+3=0}$. What is the value of b?

Proving roots of quadratic equations
If ${\displaystyle \alpha ,\beta }$ are the roots of the quadratic equation ${\displaystyle a{x}^2+bx+c=0}$, obtain the equation whose roots are ${\displaystyle \frac{1}{{\alpha }^3}\text{and}\frac{1}{{\beta }^3}}$.
If, in the above equation ${\displaystyle \alpha {\beta }^2=1}$, prove that ${\displaystyle a^3+c^3+abc=0}$.

Prove this has real roots ${\displaystyle (a^2-bc)x^2+(a+b)(a-c)x+a(b-c)=0}$

S and G are roots of ${\displaystyle x^2-3x+1=0}$. Find equation whose roots are ${\displaystyle \frac{1}{S-2}}$ and ${\displaystyle \frac{1}{G-2}}$.

Region of the coefficients of a quadratic equation that cause the roots of it to be in the unit disk
From Simon Haykin's Adaptive Filter Theory: consider the characteristic equation is ${\displaystyle 1+a_1{z}^{-1}+a_2{z}^{-2}=0}$, then for the roots to be inside the unit circle (or in the unit disk), the coefficients of the quadratic must satisfy the following conditions: ${\displaystyle -1\le a_1+a_2,-1\le a_2-a_1}$, and ${\displaystyle -1\le a_2\le 1}$.
I could figure this out some of the way. For complex roots it follows that the coefficients lie in the region defined by the intersection of ${\displaystyle 4a_2>a_1^2}$ and ${\displaystyle a_2<1}$. For real roots, I could prove that ${\displaystyle \left|a_2\right|<1}$, and obviously ${\displaystyle 4a_2\le a_1^2}$. But I cannot figure out how the line boundaries of the region can be derived.

Reduce quadratic form of 3 variables to sum of 3 squares
I am dealing with following quadratic form: ${\displaystyle q:{\mathbb{R}}^3\to \mathbb{R}}$
${\displaystyle q(x,y,z)=xy+yz+xz}$

Question regarding derivation of the quadratic formula
To derive the quadratic formula, I want to find (m,n) such that
${\displaystyle a{x}^2+bx+c+p={(mx+n)}^2+c}$
Expanding the RHS reveals
${\displaystyle a{x}^2+bx+c+p=m^2{x}^2+2mnx+n^2+c.}$
Then, ${\displaystyle a=m^2,b=2mn\Rightarrow (m,n)=(\pm \sqrt{a},\pm \frac{b}{2\sqrt{a}})}$
From here, isolating and solving for x would be trivial.Would this general method be correct? I'm confused as to the significance of p since ${\displaystyle p=n^2\Rightarrow n=\pm \sqrt{p}}$ here as well.
Perhaps it's been a long day or my algebraic skills have worsened, but I just couldn't seem to work out the quadratic formula with this path of thinking; there would always be some extra term...
I'm aware that dividing the entire original equation by a simplifies things a lot, but that should just be a specific case of this general form.

Quadratic equation with complex variable
Solve the following equation with ${\displaystyle \omega \in \mathbb{C}}$ and ${\displaystyle \gamma ,{\omega }_0\in \mathbb{R}}$ for its complex roots ${\displaystyle {\omega }_1,{\omega }_2}$
${\displaystyle {\omega }^2+\gamma \omega +{\omega }_0^2=0}$.

Quadratic formula and isolating f(x)
${\displaystyle f\left(x\right)-\frac{1}{f\left(x\right)}=g\left(x\right)}$

Prove an inequality involving a root of a quadratic equation
If ${\displaystyle x=\rho }$
is a solution to: ${\displaystyle x^2+bx+c=0}$
Prove that ${\displaystyle \left|\rho \right|-1<\left|b\right|+\left|c\right|}$

I know an index of 1.18404 for the equation "${\displaystyle 1.18404={(1+i)}^4}$", but not how to solve for i.

I have to following problem:
${\displaystyle q^2+3q=40}$
The anwser is ${\displaystyle q=5}$ and ${\displaystyle q=-8}$ can someone give me the steps?

I have the quadratic equation ${\displaystyle 5x^2+96x-576=0}$
I wonder can we solve it without using formula ${\displaystyle x=\frac{-b\pm \sqrt{\mathrm{\Delta }}}{2a}}$?