Quadratic Function Real Life Examples and Applications

To find the sum of first n numbers using the sum of odd numbers and even numbers
We know that the sum of first n numbers is ${\displaystyle \frac{n(n+1)}{2}}$. So, I tried to derive it using the formula for sum of first n odd numbers and sum of first n even numbers. Here's what I did
What I tried was to split
${\displaystyle 1+2+3+\cdots +n}$
into
${\displaystyle (1+3+5+\cdots +(2n-1))+(2+4+6+\cdots +2n)}$

The value of parameter a for which ${\displaystyle \frac{a{x}^2+3x-4}{a+3x-4x^2}}$ takes all real values for ${\displaystyle x\in R}$ are:
My question is why we need to validate end points i.e. 1,7 (Refer the last part of my attempt)

The ${\displaystyle x^2+ax+b=0}$ has only one solution and this is ${\displaystyle x=-1+\frac{1}{2}}$. What are the values of a and b?

The quadratic equation ${\displaystyle y=a{x}^2+bx+c}$ has roots ${\displaystyle x=\alpha \text{and}x=\beta }$. Find an equation that has roots ${\displaystyle x={\alpha }^2\text{and}x={\beta }^2}$.

How do you solve a quadratic equation containing a negative power?
${\displaystyle n^2-4+\frac{6}{n}=0}$

What is the equation of the quadratic function whose vertex of the graph is on the x-axis and passes through the two points (1,4) and (2,8)?

Upper bound of a quadratic, given the upper bound of variable
Let p(x) be a quadratic polynomial ${\displaystyle p\left(x\right)=a{x}^2+bx+c}$ such that ${\displaystyle \left|p\left(x\right)\right|\le 1\text{for}\left|x\right|\le 1}$. Prove that ${\displaystyle |c{x}^2+bx+a|\le 2\text{for}\left|x\right|\le 1}$.

Trying to analyze degenerate case of quadratic equation
${\displaystyle a{x}^2+bx+c=0}$ when ${\displaystyle a=0,b=0,c\ne 0}$

To find the location of roots of an equation
Prove that ${\displaystyle \frac{e^{\pi }}{x-\pi }+\frac{{\pi }^e}{x-e}+\frac{e^e+{\pi }^{\pi }}{x-e-\pi }=0}$. has one real root in ${\displaystyle (e,\pi )}$ and other ${\displaystyle (\pi ,\pi +e)}$.

The sum of the values of a for which ${\displaystyle \frac{x^3-6x^2+11x-6}{x^3+x^2-10x+8}+\frac{a}{30}=0}$ does not have a real solution is
A. 1
B. 12
C. 13
D. 2

The set of all real number x such that
${\displaystyle ||3-x|-|x+2\mid \mid =5}$

The quadratic polynomial P(x) has a zero at ${\displaystyle x=2}$. The polynomial P(P(x)) has only one real zero at ${\displaystyle x=5}$. Compute P(0).

If a and c are odd ' numbers and ${\displaystyle a{x}^2+bx+c=0}$ has rational roots, where ${\displaystyle b\in I}$, then prove that one root of the equation will be independent of a and b.

Identical roots of a quadratic equation
Let ${\displaystyle f\left(x\right)=a{x}^2+bx+c}$ and ${\displaystyle g\left(x\right)=x^2+d}$.
Now consider ${\displaystyle f\left(x\right)+\lambda g\left(x\right)=0}$

I'm trying to find the values a, b and c that would validate ${\displaystyle y=a{x}^2+bx+c}$ with the following parameters:
- For ${\displaystyle x=1;y=1}$
- For ${\displaystyle x=T;y=S}$

I'm learning how to convert quadratic equations from general form to standard form, in order to make them easier to graph. We know the general form is ${\displaystyle a{x}^2+b{x}^2+c}$, and the standard form is ${\displaystyle a{(x-h)}^2+k}$. To help with the conversion, we can expand the standard form, and see that it turns into the general form. I totally get how to go from standard to general. I can easily memorize what h and k are, and use them to consistently derive standard forms.
What I'm curious about is how to, a priori, go from the general form to the standard form? Is there a way to see that ${\displaystyle a{x}^2+bx+c}$ can turn into ${\displaystyle a{(x-h)}^2+k}$ without knowing that form ahead of time? How was the form ${\displaystyle a{(x-h)}^2+k}$ discovered in the first place? How are alternate forms of equations discovered in general? I honestly wouldn't know where to begin.
I ask out of curiosity, and because I believe knowing how to go in the other direction will help really solidify this concept for me. Even if that knowledge is above my skillset at the moment, at least an overview of what kind of math is involved may supplement this concept for me.

I need to find the step size ${\displaystyle {\alpha }_k}$ when considering function ${\displaystyle Q\left(x\right)=\frac{1}{2}{x}^{T}Ax+b^{T}x}$
I know that the gradient(g) is ${\displaystyle Ax-b}$ and I should minimize the function
${\displaystyle \mathrm{\Phi }\left(\alpha \right)=Q(x^k-\alpha g\left(x^k\right))}$
I am stuck after putting the values into ${\displaystyle {\mathrm{\Phi }}_k\left(\alpha \right)}$ and even that I am not sure of doing correctly. Can you help me with the ${\displaystyle {\mathrm{\Phi }}_k\left(\alpha \right)}$ function and if possible give some directions on what to do next?

How to show if a linear function through a point passed through a curve twice?
Show that any linear function passing through P(0, 3) will meet the curve
${\displaystyle f\left(x\right)=2x^2-x-2}$ twice

How to generate factors given the b-value of the quadratics is the hypotenuse of a Pythagorean Triple?
Given two quadratics with integer coefficients,
${\displaystyle y=x^2+bx+c}$ and ${\displaystyle x^2+bx-c}$

Use the quadratic formula to derive the Cauchy-Schwarz Inequality.
Let f and g belong to ${\displaystyle L^2\left(E\right)}$. From the linearity of integration, for any number
${\displaystyle \lambda :{\lambda }^2{\int }_E{f}^2+2\lambda {\int }_{E}f\cdot g+{\int }_E{g}^2={\int }_E{(\lambda f+g)}^2\ge 0}$
From this and the quadratic formula directly derives the Cauchy-Schwarz Inequality.