State the solutions, or estimated solutions, to the quadratic equation

How to factor a quadratic function based on variables not numbers?
${\displaystyle s^2+\frac{s}{2CR}+\frac{1}{CL}=(s+a)(s+b)}$

Indetermined system
$\sum _{i=1}^N{x}_i=\sum _{i=1}^N\frac{1}{x_i}=3$
for $N>2$. The main aspect that confuses me is the general $N$ and I'm not sure how to proceed.

Simplify ${\displaystyle x^2-3x-4÷(x+2)}$

Is it true that vector spaces are defined to check if system of linear equations is solvable or not?
Explanation: Goal is to solve system of linear equations.
In matrix form: ${\displaystyle X=b}$. As ${\displaystyle A=[C_1{C}_2\dots C_n]}$, where ${\displaystyle C_n}$ is a column and ${\displaystyle x=[x_1{x}_2\dots x_n]}$.
Therefore, ${\displaystyle C_1{x}_1+c_2{x}_2+\dots +C_n{x}_n=b}$. Linear combination of column vectors produce vector b.
Because of above statement (linear combination) we choose a set of vectors that have closure under addition and scalar multiplication (closure under linear combination) and call that set of vectors a vector space. Now, if vector b lies in that set of vectors (vector space) then only system of linear equations is solvable.

Solve the problem by writing and solving a suitable system of equations.
1) If 50 pounds of tomatoes and 30 pounds of bananas cost $45 and 10 pounds of tomatoes and 20 pounds of bananas cost $16, what is the price per pound of tomatoes and bananas?

Compute $$69\sqrt{2028}-31\sqrt{2700}$$.

Technique for solving ${\displaystyle \frac{ax+b}{cx+d}=\frac{px+q}{rx+s}}$ where the sum of numerators equals the sum of denominators
I was looking up some shortcuts to solve quadratic equations. I got a technique that applies when the sum of the numerators and denominators are equal, but I am unable to understand the reasoning behind it. Here I'm showing an example:
${\displaystyle \frac{3x+4}{6x+7}=\frac{5x+6}{2x+3}}$