Quadratic Function Real Life Examples and Applications

Find all ${\displaystyle q\in \mathbb{N}}$ such that ${\displaystyle \frac{q(q+1)}{12}}$ is a Perfect square.

Find all points on the XY plane where parabola ${\displaystyle y=x^2-4px+2p^2-3}$ doesn't pass. Note that ${\displaystyle P\in \mathbb{R}}$.

Extraneous solutions without performing squaring operation
Solve in ${\displaystyle [0.2\pi ]}$
${\displaystyle \sec x+\tan x=2\cos x}$

Evalution of a function where ${\displaystyle t=x+\frac{1}{x}}$
Consider a function
${\displaystyle y=(x^3+\frac{1}{x^3})-6(x^2+\frac{1}{x^2})+3(x+\frac{1}{x})}$
defined for real ${\displaystyle x>0}$. Letting ${\displaystyle t=x+\frac{1}{x}}$ gives:
${\displaystyle y=t^3-6t^2+12}$

The equation ${\displaystyle x^2-y^2=a^2}$ changes to the form ${\displaystyle xy=c^2}$ if the co-ordinate axes rotates through an angle (keeping origin fixed)
a) ${\displaystyle \frac{\pi }{2}}$
b) ${\displaystyle -\frac{\pi }{2}}$
c) ${\displaystyle \frac{\pi }{4}}$
d) ${\displaystyle \frac{\pi }{3}}$

Suppose the quadratic ${\displaystyle x^2+ax+b}$ equals 0 when ${\displaystyle x=c}$ or ${\displaystyle x=d}$. If ${\displaystyle c^{2}d+d^{2}c=10}$, and a and b are integers, what are all possible ordered pairs (a,b)?

Suppose ${\displaystyle a^2-4b\ne 0}$. Let ${\displaystyle \alpha ,\beta }$ be the (distinct) roots of the polynomial ${\displaystyle x^2+ax+b}$
Then there is a real number c s.t.

Subtract ${\displaystyle 5y^2-6y-11}$ from ${\displaystyle 6y^2-2y+5}$

Solving simultaneous equations with multiple variables.
If ${\displaystyle {(x+a)}^2}$ is a factor of ${\displaystyle x^3+6px+k}$, show that ${\displaystyle k+2a^3=0}$
I've tried different ways to solve this, but is it even possible?

Solving quadratic equation in motion equations
Is it possible to solve this equation for t
${\displaystyle x_f=x_0+v_{0}t+t\frac{1}{2}a{t}^2\sim ?}$
It feels like its a quadratic but since x final and initial are not the same and sometimes initial is not zero any idea?

Solving quadratic inequality ${\displaystyle x^2>0}$

Solving quadratic inequalities algebraically without graphing/common-sensing the equality signs
I've just been presented an inequality at GCSE (high school) level that, solving algebraically (the method used for linear inequalities), doesn't work correctly which has caught me by surprise. Research always leads to graphed or common-sensed solutions. Can anyone explain why this doesn't work and if there's an algebraic manipulation that leads to the right answers?
${\displaystyle x^2-49>0}$
Using the quadratic identity
${\displaystyle (x+7)(x-7)>0}$
Divide both sides by ${\displaystyle (x+7)}$...
${\displaystyle x-7>0}$
${\displaystyle x>7}$
Divide both sides by ${\displaystyle (x-7)}$...
${\displaystyle x+7>0}$
${\displaystyle x\succ 7}$
Alternatively,
${\displaystyle x^2-49>0}$
${\displaystyle x^2>49}$
${\displaystyle x>\pm 7}$
The negative solution has to be
${\displaystyle x<-7}$
but I don't know how that's arrived at through algebra.

Solving ${\displaystyle 3x^2+4x-2=0}$ (mod 31)

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

Find all values of a that satisfy the inequality for all (x,y).

Find all natural numbers x and y satisfying the equality ${\displaystyle 19(x^2+y^2)=221+33xy}$

Factorize ${\displaystyle ab{x}^2-(a^2+b^2)x+ab}$

Factor completely.
${\displaystyle 3x^2+42x+39}$

Factor quadratic equation formula
${\displaystyle a{x}^2+bx+c=(dx+e)(fx+g)}$, and my goal is to find a formula to find d, e, f, and g when given a, b, and c and that also works for complex numbers