Algebra Word Problem Solutions

For the equations given below, which statement is true?
$$\begin{gathered} -<!--\; -\; -->3x-<!--\; -\; -->8=19 \\ -<!--\; -\; -->3x-<!--\; -\; -->2=25 \end{gathered}$$
The equations have the same solution because the second equation can be obtained by adding 6 to both sides of the first equation.
The equations do not have the same solution because the second equation can be obtained by adding 6 to both sides of the first equation.
The equations have the same solution because the second equation can be obtained by subtracting 19 from both sides of the first equation
The equations have the same solution because the second equation can be obtained by subtracting 6 from both sides of the first equation.

Convert 0.625 into a fraction in its simplest form

The smallest natural number is?

What number is $10$ percent of $50$?

How do you factor the expression $x^2-7x+12$?

.$222=6,333=6,444=6,555=6,666=6,777=6,888=6,999=6,$? Explain how and why?

Which of the following is true regarding the solutions to the logarithmic equation below?
$2{\log }_6<!--\; \; -->(x)=2$
$$\begin{gathered} {\log }_6<!--\; \; -->(x^2)=2 \\ {x}^2=6^2 \\ {x}^2=36 \\ x=6,-<!--\; -\; -->6 \end{gathered}$$
x=6 and x=-6 are true solutions
x=6 and x=-6 are extraneous solutions
x=6 is a true solution and x=-6 is an extraneous solution
x=6 is an extraneous solution and x=-6 is a true solution

Use the compound interest formula $A=P(1+r{)}^t$ to find the annual interest rate, r, if in 2 years an investment of $3,000 grows to $3,630

A hospital bill is estimated to be $462.00. It ends up actually costing the patient $525.00. What is the margin of error in the bill?
A. 7%,
B. 15%,
C. 9%,
D. 12%
Include an explanation with your response.

Randy has $3.55 worth of dimes and quarters in his pocket. The number of dimes is four more than twice the number of quarters. Which system of equations can be used to find how many dimes (d) and quarters (q) Randy has in his pocket?

Find the slope of any line perpendicular to the line passing through (2,2) and (3,−5)

A grasshopper coved a distance of 5 yards in 9 equal hops. How many yards did the grasshopper travel on each hop.

Which inequality matches the graph? X, Y graph. X range is negative 10 to 10, and Y range is negative 10 to 10. Dotted line on graph has positive slope and runs through negative 3, negative 8 and 1, negative 2 and 9, 10. Above line is shaded.
$$\begin{gathered} -<!--\; -\; -->2x+3y>7 \\ 2x-<!--\; -\; -->3y<7 \\ -<!--\; -\; -->3x+2y>7 \\ 3x-<!--\; -\; -->2y<7 \end{gathered}$$

Domain and range of $g(x)=\frac{3x^2+4}{x-<!--\; -\; -->2}$
for the domain, I find, $\{x|x\ne <!--\; \ne \; -->2\}$
and for the range
$y={\displaystyle \frac{3x^2+4}{x-<!--\; -\; -->2}}$ and when $x=2-<!--\; -\; -->\mathrm{ϵ<!--\; ϵ\; -->}$, $y\to <!--\; \to \; -->{\displaystyle \frac{3(2-<!--\; -\; -->\mathrm{ϵ<!--\; ϵ\; -->}{)}^2+4}{(2-<!--\; -\; -->\mathrm{ϵ<!--\; ϵ\; -->})-<!--\; -\; -->2}}={\displaystyle \frac{K}{-<!--\; -\; -->\mathrm{ϵ<!--\; ϵ\; -->}}}\to <!--\; \to \; -->-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->}$
When $x=2+\mathrm{ϵ<!--\; ϵ\; -->}$, $y\to <!--\; \to \; -->{\displaystyle \frac{3(2+\mathrm{ϵ<!--\; ϵ\; -->}{)}^2+4}{(2+\mathrm{ϵ<!--\; ϵ\; -->})-<!--\; -\; -->2}}={\displaystyle \frac{K}{\mathrm{ϵ<!--\; ϵ\; -->}}}\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$
Therefore, $\text{ran}(g(x))=(-<!--\; -\; -->\mathrm{\infty <!--\; \infty \; -->},\mathrm{\infty <!--\; \infty \; -->})$
have I done this right or am I missing something? For instance, I thought about the case when $x\to <!--\; \to \; -->\mathrm{\infty <!--\; \infty \; -->}$, but this looks like $y\to <!--\; \to \; -->1$ in this case, nothing remarkable. I don't even know if that is what we say when we have an $\frac{{\mathrm{\infty <!--\; \infty \; -->}}^2}{\mathrm{\infty <!--\; \infty \; -->}}$....Do we just say that is one? Or do we say that is $\mathrm{\infty <!--\; \infty \; -->}$ still?

A polynomial P Is given.
$P(x)=x^6-<!--\; -\; -->7x^3-<!--\; -\; -->8$
Px)= ?
(a) Find all zeros of P, real and complex. (Enter your answers as a comma-separated list. Enter all answers including repetitions.)
x=?
(b) Factor P completely.
P(x)=?

Complex Quadratic Programming Problem
I have a quadratic programming problem as follows:
$\underset{x}{min}\Vert <!--\; \Vert \; -->Ax+b{\Vert <!--\; \Vert \; -->}_2$
with no constraints. If A,x, and b are real, It would be a straight forward QP problem that can be also written as follows:
$\underset{x}{min}\frac{1}{2}{x}^{T}Jx+c^{T}x$
where $J=A^{T}A$ and $c=A^{T}b$. Such problems can be handled by Matlab. However, in my case, A, x, and b are all complex, and therefore can't be solved by Matlab (My only tool at the moment). Of course, the target function (the norm) is real.
My question: Any Idea on how to reform such a problem to get a real QP problem so I can use Matlab to solve it?

Show that $\frac{1}{{\log }_2<!--\; \; -->x}+\frac{1}{{\log }_3<!--\; \; -->x}+\cdots <!--\; \cdots \; -->+\frac{1}{{\log }_{43}<!--\; \; -->x}=\frac{1}{{\log }_{43!}<!--\; \; -->x}$.I am just not able to get it.please help.

Finding the integral closure of k[x] in $k(x)(\sqrt{f})$, where $f(x)=x^6+t{x}^5+t^2{x}^3+t\in <!--\; \in \; -->k[x]$
Let F be a field of characteristic 2. Let $k:=F(t)$, the field of rational functions in the variable t. Let $f(x)=x^6+t{x}^5+t^2{x}^3+t\in <!--\; \in \; -->k[x]$.
We want to show that the integral closure of k[x] in $k(x)(\sqrt{f})$ is $k[x][\sqrt{f}]$.
So let $\mathrm{\alpha <!--\; \alpha \; -->}=m+n\sqrt{f}\in <!--\; \in \; -->k(x)(\sqrt{f})$, where $m,n\in <!--\; \in \; -->k(x)$. Since F(t) is a field, k[x] is a PID and therefore $\mathrm{\alpha <!--\; \alpha \; -->}$ is integral over k[x] if and only if its minimal polynomial in k(x) has coefficients in k[x]. Let's then find the minimal polynomial of $\mathrm{\alpha <!--\; \alpha \; -->}$.
$\begin{array}{rl}p(y)& =(y-<!--\; -\; -->(m+n\sqrt{f}))(y-<!--\; -\; -->(m-<!--\; -\; -->n\sqrt{f}))\\ & =y^2-<!--\; -\; -->2my+(m^2-<!--\; -\; -->n^{2}f)\end{array}$
and since $m\in <!--\; \in \; -->k(x)=F(t)k(x)$ and the characteristic of F equals 2 we get that $2my=0$. Hence
$p(y)=y^2+(m^2-<!--\; -\; -->n^{2}f)$
Thus, $\mathrm{\alpha <!--\; \alpha \; -->}=m+n\sqrt{f}$ is integral over k[x] iff $m^2-<!--\; -\; -->n^{2}f\in <!--\; \in \; -->k[x]$. I' m pretty sure that I'm on the right course but I don' t know how to continue.

180 days equals how many months?

Solving for $x$ in an exponential equation
Say we the following equation
$F(x)=\frac{\exp <!--\; \; -->(a+bx)}{1+\exp <!--\; \; -->(a+bx)}$
Now we set $x=0$ and we want to solve for $a$ as a function of $F_0$.
So that, we have:
$F_0=\frac{\exp <!--\; \; -->(a)}{1+\exp <!--\; \; -->(a)}$
Can someone please walk me though the logarithmic transformation on how we achieve this end result
$a=\ln <!--\; \; -->\left(\frac{F_0}{1-<!--\; -\; -->F_0}\right)$