Algebra Word Problem Solutions

Equality of a quadratic function
Let $f:\mathbb{R}\to <!--\; \to \; -->\mathbb{R}$ an arbitrary function and $g:\mathbb{R}\to <!--\; \to \; -->\mathbb{R}$ a quadratic function with the following property:
For any m and n the equation $f(x)=mx+n$ has a solution iff the equation $g(x)=mx+n$ has a solution.
Prove that f and g are equal.

Let's say we have the general solution to $X^{\prime }=A(t)X$, where $X=(x_1,x_2{)}^T$. How do you find the general solution to the system $X^{\prime }=A(t)X+b(t)$ where $b(t)$ is a $2\times <!--\; \times \; -->1$ matrix with two polynomials as entries. How do you find the particular solution?

${\displaystyle x^2+9\le 0}$

Write an equation of a line going through ${\displaystyle (\frac{1}{8},-1)}$ and parallel to the line 8x−9y=6

Find the slope of a line parallel to x=4y+7

Find the nth term of the geometric sequence 360,180,90,45

Value Sets miss an AP
For any polynomial $p\in <!--\; \in \; -->\mathbb{Z}[X]$, the set $\{p(n):n\in <!--\; \in \; -->Z\}$ is called the value set of p.
Problem:- Let $p\in <!--\; \in \; -->Z[X]$ be such that deg $p>1.$ Then there exists an infinite arithmetic sequence none of who terms can be expressed as p(x) for some $x\in <!--\; \in \; -->Z.$
Progress:-
For this problem, I first took example which was $P(x)=x^2-<!--\; -\; -->1$ then we get the value set as {0,3,4,15,24,…}. Since, $x^2-<!--\; -\; -->1\equiv <!--\; \equiv \; -->-<!--\; -\; -->1,0\mathrm{mod}4.$. So clearly the AP 2,6,10,… never come.
I assumed the contrary. Hence for any n,d there will exist a x such that $P(x)\equiv <!--\; \equiv \; -->n\mathrm{mod}d.$. If there doesn't then we can select the AP $n,n+d,n+2d,\dots <!--\; \dots \; -->$
Now we know that when $p\in <!--\; \in \; -->\mathbb{[}X]$, and $m\equiv <!--\; \equiv \; -->n\mathrm{mod}d⟹<!--\; ⟹\; -->P(m)\equiv <!--\; \equiv \; -->P(n)\mathrm{mod}d.$. (*)
So, we can say that $p(n),p(n+1),\dots <!--\; \dots \; -->,p(n+d-<!--\; -\; -->1)$ forms a complete residue set, because if it doesn't then by PHP, there will be an $y\in <!--\; \in \; -->1,2,\dots <!--\; \dots \; -->d$ not being a residue in p(x), when $n\le <!--\; \le \; -->x\le <!--\; \le \; -->n+d-<!--\; -\; -->1.$
Now, take any d consecutive numbers,say $l,l+1,\dots <!--\; \dots \; -->,l+d-<!--\; -\; -->1,$ by (*), residues mod d of $\{P(l),\dots <!--\; \dots \; -->,P(l+d+1)\}=\{p(n),p(n+1),\dots <!--\; \dots \; -->,p(n+d-<!--\; -\; -->1)\}.$
This is what I have got till now.

Find the value of $\frac{\sin <!--\; \; -->24x}{\cos <!--\; \; -->18x\cos <!--\; \; -->6x}$
Given $\tan <!--\; \; -->x=\sqrt{5}-<!--\; -\; -->2$, find the value of
$\frac{\sin <!--\; \; -->24x}{\cos <!--\; \; -->18x\cos <!--\; \; -->6x}.$
Now I wrote numerator as $\sin <!--\; \; -->(18+6)x$ and I used identity $\sin <!--\; \; -->(A+B)$, after that I reduced problem to $\tan <!--\; \; -->18x+\tan <!--\; \; -->6x$
How do I proceed? Thanks

Arithmetic progression and polynomials
Suppose the quadratic polynomial
$p(x)=a{x}^2+bx+c$
has positive coefficients a,b,c such that these are in AP in the given order. If m and n are the integer zeros of the polynomial then what is the value of $m+n+mn$? I have tried quadratic formula but ain't getting the answer.

Critique reasoning
Molly is wrapping parts of sandwiches to sell at her sandwich cart . She cuts each sandwich in fourths and then wraps each fourth separately. She says that she wrapped 16 fourths, so she wrapped 16 hole sandwiches. What was Molly's mistake?

Dividing logarithms without using a calculator
The problem I have is: $\frac{\log <!--\; \; -->16+\log <!--\; \; -->25-<!--\; -\; -->\log <!--\; \; -->36}{\log <!--\; \; -->10-<!--\; -\; -->\log <!--\; \; -->3}$
(log is base 10 here)
I have the answer as 2 but no idea how to reach it..
I need to work this out without the use of a calculator but I can't get my head round it. I know that
$\log <!--\; \; -->(16)=\log <!--\; \; -->(4^2)=2(\log <!--\; \; -->4)$ $\log <!--\; \; -->25=2(\log <!--\; \; -->5)$
I can add \logs together when they all are a power of the same number e.g.
$\log <!--\; \; -->(64)+\log <!--\; \; -->(32)=\log <!--\; \; -->(2^6)+\log <!--\; \; -->(2^5)=6(\log <!--\; \; -->2)+5(\log <!--\; \; -->2)=11(\log <!--\; \; -->2)$
I might just be over thinking it or over complicating it but I'd really appreciate some help here. Thanks.

Find three consecutive even integers whose sum is 244

Joe flies a plane 300 miles in 2 hours to a destination with the wind. On the return trip against the wind, he only flies 270 miles in 2 hours. Assuming wind speed is constant, what's the wind speed?

Find parametric equations and symmetric equations for the line. The line through the points (1,3,2) and (-4,3,0)

Find the 19th term in the sequence −14,−7,0,7,14,...

How to solve this recurrence relation $f(x,0)=3x\text{ and }f(x,n+1)=f(f(x,n),n)$ and $f(x,0)=3x\text{ and }f(x,n+1)=f(f(x,n),n)$?

The cost of a school banquet is $65 plus $13 for each person attending. How do you determine the linear equation that models this problem and what is the cost for 81 people?

Logarithmic inequality: ${\log }_{1/3}^2<!--\; \; -->(x^2-<!--\; -\; -->3x+2)-<!--\; -\; -->{\log }_{1/3}<!--\; \; -->(x-<!--\; -\; -->1)>{\log }_{1/3}<!--\; \; -->(x-<!--\; -\; -->2)+6$

Write an nth term rule for r=6 and ${\displaystyle a_3=72}$

How to compute the derivative of ${\sqrt{x}}^{\sqrt{x}}$?
I know have the final answer and know I need to use the natural log but I'm confused about why that is.
Could someone walk through it step by step?