Linear, exponential, logarithmic, polynomial, rational, and trigonometric functions, conic sections, binomial, surds, graphs and transformations of graphs, equation- and system-solving, and other symbolic-manipulation topics.

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0
votes
4answers
37 views

How to isolate this variable?

$$m \cdot (a - b) + c = a$$ I can't figure out how to isolate $a$ in the equation above. I can't seem to detach $m$ so that $a$ can be left alone. What steps are required to isolate $a$?
1
vote
2answers
84 views

Am I using sandwich theorem incorrectly?

I saw this question and wondered how OP of that question was able to do : $$0<\sin x+1<2$$ this $$\frac 0{|x|}<\frac{\sin x+1}{|x|}<\frac 2{|x|}$$ and when $x\to \infty$ he got the limit ...
0
votes
1answer
12 views

Algebra for a polytropic process, thermo

Polytropic process equation $p_1v_1^n=p_2v_2^n$ $pv=RT$ Where R is a constant, (Ideal gas law) How do you obtain the expression: $\frac{T_2}{T_1}=\left (\frac{p_2}{p_1}\right)^\frac{n-1}{n}$ ...
2
votes
2answers
47 views

To show that the variables in the system are same in magnitude

I am stuck with this interesting problem, If for non-negative integers $a, b, \text{and} c$, $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are both integers then ...
0
votes
1answer
49 views

How do I evaluate $\displaystyle\prod_{r=1}^{\infty }\left (1-\frac{1}{\sqrt {r+1}}\right)$?

I am not being able to find the specific product $\prod_{r=1}^{k} \left(1-\frac{1}{\sqrt {r+1}}\right)$ so to evaluate the given problem when $k \to \infty $.
0
votes
4answers
44 views

Difference between squares and roots??? [duplicate]

Why does this happen?? $$ y = \sqrt9 \implies y=3$$ $$ y^2 = 9 \implies y=+3,-3 $$ While both equations are in same sense.
0
votes
2answers
19 views

Calculating selling price given margin + tax for tax inclusive pricing

Given the following: Cost price of an item is 100.00 (NOT including tax) Margin of 10% Tax of 8% What is the formula for finding the selling price that will meet these requirements? The final ...
1
vote
0answers
23 views

A system of equations to find highest possible outcome

I am trying to solve a really simple system of equations, but its been about 10 years since Ive done this stuff. Can someone explain how to choose which stocks to buy based on cost, projected return, ...
0
votes
4answers
79 views

Prove that there is no term independent of $x$ in the binomial expansion of $\left(x-\frac 1x\right)^{19}$

I am dealing with a fairly simple question but I'm struggling a bit to come up with a formal demonstration on why the binomial expansion of $\left(x-\frac 1x\right)^{19}$ doesn't have a term ...
3
votes
2answers
63 views

$\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1$. Find $(x+y)$.

We know that $\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1$. Find the expression $(x+y)$. My work so far: $$\left(x+\sqrt{x^2+1}\right)\left(y+\sqrt{y^2+1}\right)=1$$ ...
0
votes
0answers
34 views

Number theory/calculus/algebra etc. equivalents of Euclid's Elements?

Anybody know any books that tackle mathematical topics in a deductive, axiomatic structure akin to Euclid's Elements? Thanks.
0
votes
1answer
80 views

Expand $(a-b)^3$ without formulas

When I solve $(a-b)^2$ I take $aa-ab-ab+bb$ I do not use formulas at all because I only forget them. To solve the above example all i do is to multiply one variable or constant at the time but when I ...
0
votes
1answer
40 views

What is the name of the given formula

Could anyone say what is the name of the given formula?
3
votes
2answers
63 views

Sum of real values of $x$ satisfying the equation $(x^2-5x+5)^{x^2+4x-60}=1$

I have this equation from this paper (Q.63) Find the sum of all real values of $x$ satisfying the equation-$(x^2-5x+5)^{x^2+4x-60}=1$. My attempt- ...
1
vote
1answer
43 views

The problem of surds and indices [closed]

If $$ \frac 12 \times \left(\sqrt[3] a + \sqrt[3] b + \sqrt[3] c\right) = \frac 1{\sqrt[3] 3 - 1}\;\;\;(a>b>c)$$ then what is $a - 2b - 3c $ ?
0
votes
2answers
89 views

Where would this be on a range? ($0$ to $100\%$) Algebra [closed]

Ok, we have two expressions: $30x + 1500$ $19x + 9000$ The first thing to note is, the first expression is $100\%$ and the second expression is $0\%$. The second thing is to note in all of ...
2
votes
1answer
40 views

Understanding Why Partial Fractions Works [duplicate]

I was wondering why, not how, partial fractions work the way we are normally taught to do. To be specific: We are told that, when we have a second degree expression on the bottom that can't be ...
1
vote
2answers
30 views

Given the volume of a stack of cubes, find the number of cubes

I was trying to solve a code challenge but could not wrap my head around the math. Your task is to construct a building which will be a pile of n cubes. The cube at the bottom will have a volume ...
0
votes
2answers
37 views

Finding Launch Angle for Two Projectile Collision

I'm trying to figure out the general equation for calculating the launch angle of one projectile required when trying to find a collision between that projectile and another. For the equation I've ...
1
vote
1answer
21 views

Find the positive difference of all possible values

This problem is quite challenging to me. It is highly appreciated if someone can help me with it or give me an hint. Thank you very much! Find the positive difference of all possible values of ...
0
votes
0answers
24 views

Showing Left Side to Right Side.

Let $\mathbf x$ is a $(p\times 1)$ vector, $\mathbf\mu_1$ is a $(p\times 1)$ vector, $\mathbf\mu_2$ is a $(p\times 1)$ vector, and $\Sigma$ is a $(p\times p)$ matrix. Now I have to show ...
0
votes
1answer
396 views

Variation of parameters formula with complex imaginary roots

I am needing to use the Variation of parameters formula to solve a second order non-homogeneous equation. I have used this before however i now have an equation with complex imaginary roots My second ...
0
votes
3answers
116 views

How to solve the equation $ x^{13} = 1$ by radicals?

Is there any elementary way to solve the equation $ x^{13}= 1 $ by means of radicals? If not, how to get all the solutions? Remark: The transcendental form of the solution by means of sines and ...
0
votes
0answers
15 views

Graph method for representing multiplicity of solutions

Inspired by Degenerate zeroes Fundamental theorem of Algebra In Cartesian coordinates if a root has a multiplicity of even then it "bounces" back up from the $x$-axis, if the root has a multiplicity ...
0
votes
1answer
50 views

Function Question: Where does the $3$ come from?

Can someone explain me how did ''3'' appear here ?
-1
votes
1answer
16 views

Working through summations to show equation

Given equation 1: $$E = \sum_{k=1}^N \tau x_k g(\frac{n_k}{\tau}) + \sum_{k=1}^N n_kh(\frac{n_k}{\tau})$$ equation 2: $$E = \frac{1}{2}\gamma X^2 + \epsilon \sum_{k=1}^N |n_k| ...
1
vote
5answers
121 views

Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer)

Show that $n^2+11n+2$ is not divisible by $113^2$ ( n is integer) It's obvious that if we show $113$ doesn't divide $n^2+11n+2$ we are done...
-1
votes
2answers
24 views

show equivalence of forumulas

Give Function A: $$ \frac{1}{2} \gamma X^2 - \frac{1}{2}\gamma \sum_{i=1}^N n_k^2 $$ and Function B: $$ \epsilon \sum_{i=1}^N |n_i| + \frac{\eta}{\tau} \sum_{i=1}^N {n_i}^2$$ Can you show that ...
0
votes
3answers
114 views

Between which integers is $-3 + \sqrt 5$ situated? [closed]

I can't understand how to solve this question . I'll be thankful if you help me. Please solve it and post your complete solution & a little description of what you did. Between which integers ...
1
vote
1answer
105 views

Convergent & Cauchy Sequence related prove

(1) Consider the two convergent sequences $\{a_n\}$and $\{b_n\}$ such that $$\{a_n\}\to a$$ and $$\{b_n\}\to b$$ for $n\to\infty$. Prove that $$\{a_n+b_n\}\to a + b$$ for $n\to\infty$. (2) Prove ...
2
votes
1answer
58 views

How to graph $\frac{(x+3)}{(x+1)}$?

I'm not looking for plotting values. I mean something which can be done without it. For example, for the function $f(x) = mx + b$, we can use the slope $m$ and the y-intercept $b$ to easily graph the ...
4
votes
3answers
103 views

Why is $5\tan(54^\circ) = \sqrt{25 + 10\sqrt{5}}$ and $\tan\left(\frac{\pi}{5}\right) = \sqrt{5 - 2\sqrt{5}}$?

On the Wikipedia Page about Pentagons, I noticed a statement in their work saying that $\sqrt{25+10\sqrt{5}}=5\tan(54^{\circ})$ and $\sqrt{5-2\sqrt{5}}=\tan(\frac {\pi}{5})$ My question is: How would ...
0
votes
3answers
44 views

What is the logic behind the method used to shift a graph horizontally? [duplicate]

I understand why we add or subtract from $f(x)$ to move the graph vertically. To move it up, we move all the values up by adding whatever constant value we want. However, when it comes to horizontal ...
18
votes
2answers
2k views

Significance of $\displaystyle\sqrt[n]{a^n} $?

There is a formula given in my module: $$ \sqrt[n]{a^n} = a \text{ if $n$ is odd } $$ $$ \sqrt[n]{a^n} = |a| \text{ if $n$ is even } $$ I don't really understand the differences between them, ...
-2
votes
0answers
30 views

How to get the value a in a expression that have log function? [closed]

There is an inequality ln(2a+1)>a Could someone show me step by step, how to get the range of a? Thanks~
3
votes
2answers
25 views

Precalc Optimization?

I need help with an optimization problem. I have a rectangle space being fenced. Three sides are fenced with a material costing 4 dollars and the last side costs 16 dollars. I was given that the area ...
0
votes
1answer
40 views

Ratio & Mixture

In what ratio $3$ solutions (of milk and water $A,B,C$ ) are to be mixed to get a resultant solution $1:1$ ratio milk and water? In solution $A$ milk:water $=2:3$, in solution $B$ milk:water $=1:3$, ...
0
votes
0answers
38 views

Not understanding a question for a mathematical model using functions

You work forty hours a week at a furniture store. You receive $\$220$ as a weekly salary, plus a $3\%$ commission on sales over $5000$ dollars. Create a function to calculate the total amount you ...
2
votes
1answer
48 views

Inverse of almost quadratic functions

Hi all first question here, I'm currently trying to invert the following two (two since the $\pm$ sign) functions: $y = x^2 \pm \sqrt{(ax + b \sin{\theta})^2 + (b\cos{\theta})^2}$ Where a,b, ...
0
votes
2answers
1k views

How to combine an amount of money with the compound interest function?

Tommy has some money at home from his graduation modeled by the function $h(x)=350$. He read about a bank that has savings accounts that accrue interest according to the function $s(x)= 1.04 ...
11
votes
1answer
563 views

Series sum formula

Is there any general formula to sum following series: $$S = 1^1 + 2^2 + 3^3 + \dotsb+(n - 1)^{n - 1} + n^n, n \in N$$ I mean for $S = f(n)$, is there a formula to compute $f(n)$?
4
votes
1answer
34 views

A projectile is launched with a velocity of 30 m/s at 40° to the ground. To two decimal places, what is its horizontal velocity in m/s?

A projectile is launched with a velocity of $30 \, m/s$ at $40^\circ$ to the ground. To two decimal places, what is its horizontal velocity in $m/s$? A ball is thrown downward from the top of a ...
7
votes
3answers
1k views

Showing that $\max\{f+g\} \leq \max f + \max g$

Given real-valued continuous functions $f, g$, is the following (and why?) inequality true? $$\max \{f + g \} \leq \max f + \max g$$ Can someone give me a proof? I suspect the min is the reverse ...
0
votes
3answers
28 views

shortcut technique to solve algebric problem

If we multiply three consecutive numbers, we get 120. what the summation of those numbers? Is there any shortcut way of solving this problem without doing much ...
4
votes
4answers
100 views

Find the two square roots of $i$

I have this question I am stumped upon for my Test-Review: Write $i$ as a complex number in polar form. Use the result and DeMoivre's Theorem to find the square roots of $i$. I got the first ...
0
votes
0answers
37 views

Finding $x^2$ and $y^2$ of hyperbola

Currently, I am trying to the $x^2$ and $y^2$ of a hyperbola. I have the vertices at $(-1, -1)$ $(5, -1)$ I have the focus at $(-4, -1)$ $(8, -1)$ I know that the distance between two vertices ...
5
votes
1answer
43 views

Prove : $2(x^3+y^3+z^3)+3xyz \ge 3(x^2y+y^2z+z^2x)$ with $x,y,z \gt 0$

At first I tried to divide both side by $xyz$, the inequality became: $$2\sum {\frac{x^2}{yz}}+3 \ge 3\sum{\frac xy}$$ Let $$\frac xy = a;\frac yz = b;\frac zx = c;$$ So all we have to prove is ...
1
vote
2answers
36 views

Dividing an integer into a fixed number of integers

What is the formula for dividing an integer into a fixed number of integers where the greatest distance between consecutive integers is 1. Dividing 10 into 4 integers we can get: ...
0
votes
2answers
62 views

When can I simplify an equation?

Suppose I have a few ecuations: $$\cos^2(x) = \sin(x)\cos(x) \Rightarrow cos(x) = sin(x) $$ $$ x^2 + 3x \ge 2x \Rightarrow x(x+3) \ge 2x \Rightarrow x+3 \ge 2$$ Which of them are true and ...
-2
votes
0answers
26 views

Exercise in relative frequency of buying a coffee

Pete, Douglas and Surya buy each other rounds of coffee when they are in the office. They want to do so at a frequency that is fair, proportionate to each individual’s attendance and the attendance ...