Solve the quadratic by using the quadratic formulax^2 = 7x + 9

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 ${\displaystyle a^3+c^3+abc=0}$.

Find the vertex, the axis of symmetry, and the y-intercept of the graph.

How can we make it easy to solve quadratics without the Quadratic Formula?
For example,
${\displaystyle 5x^2-7x-2=0}$
Then, we only need to find is when
${\displaystyle 5x^2-7x=2}$
${\displaystyle x(5x-7)=2}$
Since that when we get ${\displaystyle 2-2=0}$.
Is there any way we can find them easily without solving quadratics using formula?

What is the complete factorization of ${\displaystyle 6x^4+12x^2}$?

System of two quadratics equation, P(x) and Q(x)
If ${\displaystyle P\left(x\right)=a{x}^2+bx+c\text{and}Q\left(x\right)=–a{x}^2+dx+c,ac\ne 0}$, then the equation P(x). ${\displaystyle Q\left(x\right)=0}$ has
(A) Exactly two real roots
(B) At least two real roots
(C) Exactly four real roots
(D) No real roots

Fill out the information below for the following quadratic function:
${\displaystyle f\left(x\right)=x^2-4x-5}$
Complete the table below:
$$\begin{array}{|cc|}\hline x& f(x)\\ 0& \\ 1& \\ 2& \\ 3& \\ 4& \\ \hline\end{array}$$
Write the factored form of the function
${\displaystyle f\left(x\right)=?}$

${\displaystyle 36n^2+102n-84}$