Master Algebra 1 Word Problems with our Comprehensive Practice

Rewriting a quadratic Matrix equation as a quadratic vector equation
Consider the set of $N\times <!--\; \times \; -->N$ matrices $\{W_i{\}}_{i=1}^{i=L}$, set of $N\times <!--\; \times \; -->1$ vectors $\{g_i{\}}_{i=1}^{i=L}$ and $\{h_i{\}}_{i=1}^{i=L}$. Now consider the following sum
$\begin{array}{r}S=\underset{i}{\sum <!--\; \sum \; -->}\underset{j}{\sum <!--\; \sum \; -->}{g}_i^H{W}_i{h}_i{h}_j^H{W}_j^H{g}_j\end{array}$
where the summation runs through all L for all i,j. Clearly, this equation is quadratic in the matrix variables $\{W_i{\}}_{i=1}^{i=L}$. Now define the column vector
$\begin{array}{r}w=\left[\begin{array}{c}\mathrm{vec}<!--\; \; -->(W_1)\\ \mathrm{vec}<!--\; \; -->(W_2)\\ ⋮<!--\; ⋮\; -->\\ \mathrm{vec}<!--\; \; -->(W_L)\end{array}\right]\end{array}$
where for a matrix A, vec(A) denotes the column vector containing the columns of A starting from the first column. The question is, can we write
$\begin{array}{r}S=w^{H}Qw\end{array}$
where Q is a matrix which is a function of $\{g_i{\}}_{i=1}^{i=L}$ and $\{h_i{\}}_{i=1}^{i=L}$. If so, what is the structure of Q?

Find the common ratio if geometric sequence has first term 54 and 4th term 2

Find the slope of a line parallel to the graph y-3=0

Find the first five terms of the arithmetic sequence given ${\displaystyle a_1=2}$ and d=13

Find the slope of the line perpendicular to ${\displaystyle y=\frac{1}{3}x-23}$

Find the slope that is perpendicular to the line ${\displaystyle y=-\frac{2}{3}x-5}$

Find the sum of a 12–term arithmetic sequence where the last term is 13 and the common difference is –10

In the coordinate plane, n passes through the points (-1,4) and (5,2) and m passes through the points (2,1) and (4,y). For what value of y is n perpendicular to m?

Let $A=[0,1),B=(0,1],C=(0,1)$
Prove A is of cardinality equal to B by using Glueing Lemma to 'glue together' the identity function idC and some appropriate bijective function.

How can we integrate a function for surface area of revolution?
$S=2\mathrm{\pi <!--\; \pi \; -->}{\int <!--\; \int \; -->}_0^8(-<!--\; -\; -->\frac{1}{24}{x}^2+\frac{2}{3}x)\sqrt{1+(-<!--\; -\; -->\frac{1}{12}x+\frac{2}{3}{)}^2}dx$
This is the equation (with my values plugged in) for surface area of revolution which I found online and I understand the derivation, thus I wanted to use it for a math project I am currently working on. I attempted to use the quadratic function (outside the radical) and inside the radical is the first derivative squared. However, I am stuck in evaluating the integral. I tried u substitution where u is equal to $-<!--\; -\; -->1/12x+2/3$ but was unable to finish it.
Is trigonometry required and does the u substitution I attempted not work in this case? What is the Surface area (answer to the integral)?

Find the common ratio of 136,68,34,17,...

Write the equation of a line passing through (-6, 2) and perpendicular to y=-3/5x+6

What 2 odd numbers added together equal an odd number?

What is the formula for the sequence ${\displaystyle 2,-1,4,-7,10,-13,16,...}$ ?

Find ${\displaystyle a_{12}}$ given 8, 3, -2, ...

How to evaluate a function with a quadratic argument or higher? How to proceed?
example if I have a function of the form
$$f(2x+5)=4x-<!--\; -\; -->3$$
and I am asked to evaluate
f(8)
I make
$$t=2x+5$$
from where
$$x=(t-<!--\; -\; -->5)/2$$
then
$$f(t)=4((t-<!--\; -\; -->5)/2)-<!--\; -\; -->3$$
I take
$$t=8$$
and that's it.**
Doubts
but how do I do it here, if I do the above I get a quadratic equation or a cubic one
a) $f(x^2+1/x^2)=x+1/x$
find
$$f(4)=?$$
$x>0$ b)
$$f(x^3+1/x^3)=x+1/x$$
find
$$f(4)=?$$
$x>0$

Does a pointwisely convergent sequence of quadratic functions pointwisely converges to a quadratic one?
Let $\{f_j{\}}_{j=1}^{\mathrm{\infty <!--\; \infty \; -->}}$ be a sequence of functions $f_j:{\mathbb{R}}^n\to <!--\; \to \; -->\mathbb{R}$ quadratically parameterized as $f_j(x)=x^T{P}_{j}x$ for a symmetric matrix $P_j\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$. Let $f_j$ pointwisely converges to a function f.
Then, can we say that f is also quadratic so that $f(x)=x^{T}Px$ for some symmetric $P\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$?
I was particularly interested in the case: fj is monotonically increasing and bounded by a quadratic function $g(x)=x^{T}Qx$:
$$f_1(x)\le <!--\; \le \; -->\cdots <!--\; \cdots \; -->\le <!--\; \le \; -->f_j(x)\le <!--\; \le \; -->f_{j+1}(x)\le <!--\; \le \; -->\cdots <!--\; \cdots \; -->\le <!--\; \le \; -->g(x)<\mathrm{\infty <!--\; \infty \; -->}\mathrm{\forall <!--\; \forall \; -->}x\in <!--\; \in \; -->{\mathbb{R}}^n.$$
In that case, by MCT, $f_j\to <!--\; \to \; -->f$ pointwisely for some f, and we also have $P_j\to <!--\; \to \; -->P$ for some symmetric $P\in <!--\; \in \; -->{\mathbb{R}}^{n\times <!--\; \times \; -->n}$, by MCT again, due to
$$P_1\le <!--\; \le \; -->\cdots <!--\; \cdots \; -->\le <!--\; \le \; -->P_j\le <!--\; \le \; -->P_{j+1}\le <!--\; \le \; -->\cdots <!--\; \cdots \; -->\le <!--\; \le \; -->Q.$$
In this case, $f(x)=x^{T}Px$ is true since $x^T{P}_{j}x\to <!--\; \to \; -->x^{T}Px$ for all $x\in <!--\; \in \; -->{\mathbb{R}}^n$. Moreover, if $f(x)=x^{T}Px$ is true, then it is trivially continuous, so that the convergence is uniform on any compact $\mathrm{\Omega <!--\; \Omega \; -->}\subset <!--\; \subset \; -->{\mathbb{R}}^n$ by Dini's theorem.
However, I can't see whether $f(x)=x^{T}Px$ is true or not in general case without monotonicity.

Find the value or expression for $\mathrm{\theta <!--\; \theta \; -->}$...
$\sin <!--\; \; -->(55)=\cos <!--\; \; -->(\mathrm{\theta <!--\; \theta \; -->})$

Solutions for a system of congruence equations
$$\{\begin{array}{l}x\equiv <!--\; \equiv \; -->7(\mathrm{mod}15)\\ x\equiv <!--\; \equiv \; -->14(\mathrm{mod}33)\end{array}$$

What is the 5th term of the sequence if the first term of a geometric sequence is 40, and the common ratio is 0.5