Master Algebra 1 Word Problems with our Comprehensive Practice

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.

Use the following generating function to create a sequence using ${\displaystyle n=0,1,2,3,4}$
${\displaystyle a_n=2^{n+1}}$

Use definition to find derivative of the following function at ${\displaystyle z=3-2i}$
${\displaystyle f\left(z\right)=3^2-2iz+6}$

Use the approach in Gauss's Problem and find the following sums of arithmetic sequences.
${\displaystyle 293+290+287+284+..+2}$

Unsure of reasoning behind a "shortcut" method to solve
${\displaystyle {(15y+24)}^2=12(20y^2+64y+51)}$

Two factory plants are making TV panels. Yesterday, Plant A produced 7000 fewer panels than Plant B did. Two percent of the panels from Plant A and 5% of the panels from Plant B were defective. How many panels did Plant B produce, if the two plants together produced 1260 defective panels?

To rent a certain meeting room, a college charges a reservation fee of $32 and an additional fee of $8.80 per hour. The film club wants to spend less than $76.00 on renting the meeting room.
What are the possible amounts of time for which they could rent the meeting room? Use t for the number of hours the meeting room is rented, and solve for t.

The value of a given that the cubic equation ${\displaystyle x^3+2ax+2=0}$ and the biquadratic equation ${\displaystyle x^4+2a{x}^2+1=0}$ have a common root.

The sum of ${\displaystyle 8^x}$ and its reciprocal Is 18 by substituting the value of x. My question is, is there a way to solve this within 1 minute? Like a clever shortcut or you just need to brute force this to find the answer?

For positive integers x, y, and positive real k, if ${\displaystyle \frac{x+1}{y+k}=\frac{y}{x}}$, then ${\displaystyle k\ge 1}$

For how many integer values of m parabola ${\displaystyle y=(m-2)x^2+12x+m+3}$ passes through only three quadrants?
1) 0
2) 7
3) 12
4) infinity

Finding the range of p such that ${\displaystyle p=3{\cos }^{2}x+4\sin x}$

Finding sign of leading coefficient of a quadratic equation
In a given quadratic equation ${\displaystyle f\left(x\right)=a{x}^2+bx+c}$ if ${\displaystyle f(-1)\succ 4,f\left(1\right)<0}$ and ${\displaystyle f\left(3\right)>5}$, then how can I find the sign of a? Answer in the textbook: ${\displaystyle a>0}$.

Finding inverse function of ${\displaystyle y=\sqrt{x+\sqrt{2x-1}}}$