Quadratic Function Real Life Examples and Applications

Find the inverse of ${\displaystyle x^2+4x+1,[-2,0)}$

Find or disprove the existence of a minimal quadratically closed field extension of ${\displaystyle Z_2}$ I.e.
${\displaystyle \mathrm{\forall }b,c\in S,\mathrm{\exists }x\in S,x^2+bx+c=0}$
Note: Because S is a field we can reduce every polynomial to a monic one by multiplying by ${\displaystyle a^{-1}}$.

Find all values of b for which the equations ${\displaystyle 1988x^2+bx+8891=0}$ and ${\displaystyle 8891x^2+bx+1988=0}$ have a common root.

How do we get d in terms of a here:
${\displaystyle 6{\left(d–a\right)}^2=ad}$

Moving a point along a polynomial curve by certain arc length
${\displaystyle L={\int }_n^m\sqrt{1+{(b+2ax)}^2}dx}$

Let a, b and c be odd positive integers . Show that the quadratic equation ${\displaystyle a{x}^2+bx+c=0}$ has no rational solution.

Let a, b, c are positive numbers. Prove that ${\displaystyle \frac{a^2}{{(a+b)}^2}\ge \frac{4a^2-b^2-bc+7ca}{4{(a+b+c)}^2}}$

Let a, b, ${\displaystyle c\in \mathbb{N}}$ and Quadratic equations ${\displaystyle x^2-2ax+b=0,x^2-2bx+c=0,x^2-2cx+a=0,,}$ has positive integer roots. Find a, b, c.

Let a,b,c be real numbers, ${\displaystyle a\ne 0}$. If a is a root of ${\displaystyle a^2{x}^2+bx+c=0,\beta }$ is the root of ${\displaystyle a^2{x}^2-bx-c=0}$ and ${\displaystyle 0<\alpha <\beta }$, then the equation ${\displaystyle a^2{x}^2+2bx+2c=0}$ has a root ${\displaystyle \gamma }$ that always satisfies.

Is there an easier way to solve the given problem?
If ${\displaystyle x+2+\sqrt{3}i=0}$ then find the value of ${\displaystyle 2x^4+3x^3-x^2-15x+36}$.

Is ${\displaystyle y^2+f\left(y\right)b+c}$ a quadratic equation?

Is ${\displaystyle a{x}^2+bx=0}$ considered a quadratic equation? Or is it linear, since it simplifies to ${\displaystyle ax+b=0?}$

Find the inverse of the function on the given domain
${\displaystyle f\left(x\right)={(x-10)}^2,[10,\mathrm{\infty })}$
${\displaystyle f^{-1}\left(x\right)=}$ .....
Note: There is a sample student explantion given in the feedback to this question.

Find the general solution of:
${\displaystyle {\sin }^{2}x{\cos }^{2}x+\sin x\cos x-1=0}$

Find the integer pair (a,b) that satisfies this constraint: ${\displaystyle x^2-ax-b=0}$ has integer solutions and ${\displaystyle x^2+ax+b=0}$ has integer solutions. The pair (5,6) satisfies this because ${\displaystyle x^2-5x-6=0}$ has integer solutions i.e. ${\displaystyle x=\{-1,6\}}$ and ${\displaystyle x^2+5x+6=0}$ has integer solutions i.e. ${\displaystyle x=\{-3,-2\}}$.
If there is no general form can you help me to at least find another pair of integers that satisfies this constraint other than (5,6)? Thanks

Find the interval in which a lies such that the roots of the equation ${\displaystyle (a+1)x^2-3ax+4a}$, is greater than 1.

Find p and q such that ${\displaystyle x^2+px+q<x\iff x\in (1,5)}$

Discriminant Problems
We all know that the discriminant is the part ${\displaystyle b^2-4ac}$ of the equation
${\displaystyle x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}}$
that we use to find the roots of a quadratic equation eg:
${\displaystyle a{x}^2+bx+c=0}$ or the part in a trinomial expression like ${\displaystyle a{x}^2+bx+c}$.
What I want to know is what kind of ideas we can conclude for a given inequality of a discriminant. We know that in quadratic formulas the number of real roots depends on the case whether the discriminant is <0, 0, or >0. But if given only the discriminant and the inequality of it, for examples like this:
${\displaystyle b^2-4ac=0}$
${\displaystyle b^2-4ac>0}$
${\displaystyle b^2-4ac<0}$
What can we conclude by the given information for each case? How can it help to reveal certain problems involving trinomial expressions?

Minimizing the maximum of ${\displaystyle |x^2-x-k|}$
Let k be a real number. Also, let M(k) be the maximum value of ${\displaystyle f\left(x\right)=|x^2-x-k|}$ in ${\displaystyle 0\le x\le 1}$. What are all real numbers k which minimize M(k)?

Let f(x) be a quadratic polynomial satisfying ${\displaystyle f\left(2\right)+f\left(4\right)=0}$. If unity is one root of ${\displaystyle f\left(x\right)=0}$ then find the other root.