Algebra Word Problem Solutions

${\displaystyle \frac{3}{4}}$ of the books in the library are English books and the rest are Chinese books. There are 2849 more English books than Chinese books. What is the total number of books in the library?

How to solve
${\displaystyle {\omega }^4-[{\left(\frac{eB}{m}\right)}^2+2{\omega }_0^2]{\omega }^2+{\omega }_0^4=0}$
in the simplest way

${\displaystyle -{(1+m)}^2-t-(-1+r-w)(1+r+w)}$ solve for w,m with known t, r?

${\displaystyle \left(f^{\circ }g\right)}$ given: ${\displaystyle f\left(x\right)=5x^2+6,g\left(x\right)=6x-2}$

${\displaystyle \alpha ,\beta }$ are roots of the equation ${\displaystyle 3x^2-(m-2)x+(m-5)=0}$ such that ${\displaystyle {\alpha }^5+{\beta }^5=33}$. Find the value of m.

Proving ${\displaystyle 3(1-a+a^2)(1-b+b^2)(1-c+c^2)\ge 1+abc+a^2{b}^2{c}^2}$
My task was to prove the question above over real variables.
I thought that this minor inequality should help
${\displaystyle 3(1-a+a^2)(1-b+b^2)\ge 2(1-ab+a^2{b}^2).}$
which is true.
By this inequality, the original inequality is converted to
${\displaystyle {(1-ab)}^2{(1-c)}^2+{(ab-c)}^2+abc\ge 0}$
This proves the inequality for ${\displaystyle abc\ge 0}$.
I want to prove this Inequality for ${\displaystyle abc<0}$. But I couldn't find a solution for ${\displaystyle abc<0}$.
Any extensions for ${\displaystyle abc<0}$ are thankfully accepted.

Proving Logarithmic identity
Today, I read these two new Logarithmic identity
${\displaystyle a^{{\log }_{a}m}=m}$
${\displaystyle {\log }_{a^q}{m}^p=\frac{p}{q}{\log }_{a}m}$
Both of them seems new to me,so even after solving some problems (directly) based on thesm I haven't fully understood how they holds good,Could anybody show me how to prove them ?

Prove:
${\displaystyle \underset{n\to \mathrm{\infty }}{lim}\ln \left(\frac{n+1}{n}\right)=0}$

Prove the following for the given quadratic
If the difference of the roots of ${\displaystyle x^2-px+q=0}$ is unity then prove that ${\displaystyle p^2-4q=1\text{and}{p}^2=4q^2={(1+2q)}^2}$

S(n) properties
Recently, while reading a number theory textbook for Olympiads, i came across the following property:
${\displaystyle S(n_1+n_2)\le S\left(n_1\right)+S\left(n_2\right)}$

Prove that two coefficients of a quadratic function are always positive
Let f(x) be a quadratic function of the form ${\displaystyle x^2+px+q}$ such that the equation ${\displaystyle f\left(f\left(x\right)\right)=0}$ has two equal real roots. Show that ${\displaystyle p\ge 0\text{and}q\ge 0}$.

Prove that the difference between numbers with the same sum of digits is a multiple of 9.

Prove that ${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{\mu \left(n\right)}{{10}^n}}$ is irrational

Repeating decimals in converted fractions, why these increments?
a) ${\displaystyle \frac{1}{31}=0.032258064516129\cdots }$ ., the repetend is 032258064516129 which can be broken into the numbers 032258, 064516, 129032 that are related by multiples of 2:
b) ${\displaystyle \frac{1}{13}=0.076923\cdots }$. , the repetend is 076923 which can be broken into 0769, 2307 that are related by multiplying by 3:
c) ${\displaystyle \frac{1}{17}=0.0588235294117647\cdots }$ , the repetend is 0588235294117647 which can be broken into the numbers 5882352941, 11764705882 that are related by multiplying by 2:
d) ${\displaystyle \frac{1}{19}=0.052631578947368421\cdots }$ , the repetend is 052631578947368421 which can be broken into the numbers 05263 , 15789 , 47367 , 142101 (think of the initial 1 in 142101 being removed from the 8 in the preceding 47368) that are related by multiples of 3:
e) ${\displaystyle \frac{1}{21}=0.047619\cdots }$ , the repetend is 047619 which can be broken into the numbers 0476, 1904 that are related by multiplying by 4:

Question which looks something similar to decimal expansion
Let
${\displaystyle A=\{\sum _{n=1}^{\mathrm{\infty }}\frac{a_i}{5^i}:a_i=0,1,2,3\text{ or }4\}}$
What is set A?

Quadratic inequality with Modulus of a Linear Equation : Solution Verification & Clarification Needed
${\displaystyle x^2-|5x-3|-x<2}$
For this problem I took two cases where in one case ${\displaystyle 5x-3\ge 0}$ and in second case ${\displaystyle 5x-3<0}$
1) ${\displaystyle 5x-3\ge 0}$:
2) ${\displaystyle 5x-3<0}$

Quadratic substituted into itself : wrong?
Let ${\displaystyle x^2=x+1}$ the solution is the golden ratio phi : ${\displaystyle x=\frac{1\pm \sqrt{5}}{2}}$

Quadratic with non numeric coefficents. Find where q has no real roots
${\displaystyle p{x}^2-px+3x-4}$
is for which values of p does the equation ${\displaystyle f\left(x\right)=0}$

Quadratics Distance-Speed-Time Rate Word Problem
A small plane is travelling between Windsor and Pelee Island (a distance of approx. 60 km) and is directly affected by the prevailing winds. Thus, the actual speed of the plane with respect to the ground is the speed of the plane (160 km/h) plus or minus the wind speed, w. Develop a simplified equation for the total time it takes to make a round trip if the wind speed is w.