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.

learn more… | top users | synonyms (2)

-2
votes
2answers
30 views

Ratio and Solutions

A glass contains 100 ml water. A man sips 20 ml of water from it and then adds 10 ml alcohol to it and the solution is mixed.He keeps repeating this process until the glass is empty.What will be the ...
0
votes
1answer
19 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 and in Solution C ...
6
votes
2answers
135 views

Simple limit problem with squares

I'm doing a refreshment course in math but I'm stuck with some problem. Although this problem doesn't look hard I don't know what I'm doing wrong. $$\lim_{x \to 4} ...
-1
votes
3answers
44 views

A train starts from X towards Y, which is at a distance of 55 km, at a speed of 40 km per hour.

A train starts from $X$ towards $Y$, which is at a distance of $55$ km, at a speed of $40$ km per hour. After running a certain distance, it increases its speed to $50$ km per hour and reaches $Y$ in ...
1
vote
3answers
65 views

$M=\{a+b\sqrt{2}: a,b \in \mathbb{Q} \}$ and $N=\{c+d\sqrt{3}: c,d \in \mathbb{Q}\}$. $M \cap N \subseteq \mathbb{Q}$. [closed]

Let $M=\{a+b\sqrt{2}: a,b \in \mathbb{Q} \}$ and $N=\{c+d\sqrt{3}: c,d \in \mathbb{Q}\}$. Prove $M \cap N \subseteq \mathbb{Q}$.
-1
votes
2answers
28 views

At what time and distance from Delhi will the mall train completely cross the goods train?

A goods train $158$ metres long, and traveling at the average speed of $32$ km/hr leaves Delhi at $6:00$ A.M. Another mall train $130$ metres long and traveling at the average speed of $80$ km/hr ...
1
vote
2answers
64 views

Conjugation (Group vs. Algebraic)

I am just starting to learn about groups, and the concept of conjugation came up. I was wondering what the relationship was, if any, between conjugation in the group sense and conjugation in the ...
0
votes
2answers
41 views

Supply and demand equation

I am having a hard time factoring this equation to find the equilibrium quantity and price. $11p+3x-66 =0$ and $2p^2+p-x=10$ I have gotten this far. $11p+3x-66=2p^2+p-x-10$ ...
6
votes
5answers
78 views

Prove $a^2+b^2+c^2=\frac{6}{5}$ if $a+b+c=0$ and $a^3+b^3+c^3=a^5+b^5+c^5$

if $a,b,c$ are real numbers that $a\neq0,b\neq0,c\neq0$ and $a+b+c=0$ and $$a^3+b^3+c^3=a^5+b^5+c^5$$ Prove that $a^2+b^2+c^2=\frac{6}{5}$. Things I have done: $a+b+c=0$ So ...
0
votes
1answer
44 views

How to solve algebra equation?

Today, I saw this equation on an assessment in class. Before I handed it in, I was sure to copy it down so I could ask for some help here. Alright, I had a lot of work down, and I always messed up ...
1
vote
3answers
35 views

remainder of polynomial division

The polynomial $P$ gives a remainder of $5x-7$ when divided by $x^2 -1$. Find the remainder when $P$ is divided by $x-1$. I know we can use Bezout's theorem. Thus for $x-1$ the remainder will be ...
1
vote
2answers
57 views

Upper approximation of $\mathrm{atanh}(x)$?

Is there are nice upper approximation of $\mathrm{atanh(x)}$? For example, $\ln(x)$ is nicely approximated by $x-1$ for $x$ around $1$.
1
vote
1answer
72 views

Technique for solving $ x^4 - x^3 + x - 1 = 0 $

Here's another idiotic algebra question that I can't seem to make any progress on. $$ x^4 - x^3 + x - 1 = 0 $$ I tried to make it into a quadratic: Let $ u = x^2 $, then $$ u^2 - xu + x - 1 = 0 $$ ...
0
votes
2answers
48 views

Is it possible to do this? Write a fraction as a product

I have two quantities $A$ and $B$ and I consider the fraction $$\frac{1}{A+B}$$ I would like to write the above expression as a Product, i.e. find functions $F$ and $G$ such that $$\frac{1}{A+B} = ...
1
vote
2answers
29 views

Arithmetic progression on finding first term and common difference

How to do with the question ask to find out the values of a= first term and d=common difference if the sum of the first four term is equal to three times the fourth term and the eighteenth term is 3? ...
3
votes
4answers
108 views

Why is it impossible to find natural numbers $a$ and $b$ such that $12b^2=a^2$?

This was a question in the exercises for an EdX course by Prof Starbird on Effective Thinking through Mathematics which was long over, but I am working through the course at my own pace. I feel that ...
2
votes
2answers
38 views

Finding a specific term in an expansion $(a+b)^n$ without expanding

How can I find a term within an expansion without actually expanding or using Pascal's Triangle? For example: 5th term of $$ \left(\dfrac{x}{y}-\dfrac{y}{x}\right)^8 $$
1
vote
2answers
25 views

Help with significant figures unit conversion…

So I am having trouble remembering the trick to convert from square metres etc. to square millimetres. Say I have $2\cdot{10}^{-3}m^2$ and I want to get it into millimetres. I vaguely remember ...
13
votes
10answers
4k views

Taking Calculus in a few days and I still don't know how to factorize quadratics

Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ...
1
vote
1answer
33 views

Solving simultaneous equations with `min{}` function

I have following system of m number of simultaneous equations with min{} function. These equations are symmetric as well. ...
1
vote
3answers
44 views

What are the steps to solving |3x + 1| > |2x - 7| with the given answer as $(-∞,-8)\cup(6/5,∞)$?

What are the steps to solving $|3x + 1| > |2x - 7|$ with the given answer as $(-∞,-8)\cup(6/5,∞)$? I am having difficulty with understanding inequalities with absolute value functions on both ...
2
votes
1answer
146 views

Simplify the radical

I need to simplify this radical, $\sqrt{2+e^{8t}+e^{-8t}}$ How is this done? I do not know where to go from here to simplify this further.
0
votes
1answer
53 views

To Find the height of the building

A building casts a shadow 50 feet long. A rod 4 feet tall placed near the building casts a shadow 3 inches long. Find the height of the building.
3
votes
5answers
94 views

Complex solutions to $ x^3 + 512 = 0 $

An algebra book has the exercise $$ x^3 + 512 = 0 $$ I can find the real solution easily enough with $$ x^3 = -512 $$ $$ \sqrt[3]{x^3} = \sqrt[3]{-512} $$ $$ x = -8 $$ The book also gives the ...
1
vote
0answers
55 views

Proving uniqueness

Sorry this is such a trivial question, but I'd like to check my understanding on this subject before proceeding. Suppose that we have the quantity $x+a=b$ where $x$ is a variable, and we wish to show ...
1
vote
2answers
84 views

How to simplify $\sqrt{\sqrt{5}+1} \cdot \sqrt{\sqrt{5}-1}$?

This is the original problem: $\sqrt{\sqrt{5}+1} \cdot \sqrt{\sqrt{5}-1} = x$. I'm really confused about how to solve this problem, I come as far as saying this: $\sqrt[4]{5} + \sqrt{1}\cdot ...
2
votes
2answers
48 views

Mathematics language, how to say that a specific value of $x$ is included in the functions domain?

For instance, we have the function $y=-2(x+1)^2-10$. $x=0$ is included in this functions domain. How can I say this mathematically, instead of typing out a sentence and saying that $x=0$ exists in ...
0
votes
1answer
15 views

How many points do the graphs of the following functions have on the $x$-$y$ axis? Infinite or finite?

I am stuck on this question and cannot figure it out, $$y=-4-x$$ $$y=\frac{1}{x}-x$$ The first equation is a line, so should it not have infinite points? The second equation has a restriction, $x ...
2
votes
2answers
27 views

Exponents with Logs

Could someone show work for why $e^{2\ln(x)}$ = $x^2$ ? I ran across this while solving an ODE but have completely forgotten the rules used here. I hate to ask it, but i'd rather ask it this once than ...
-2
votes
2answers
47 views

Hey guys. Given the graph below, find the equation of the transformed parent function. [closed]

It would be great if there is a detailed explanation. Also, is there a standard method I can use to answer all kinds of graphs including exponents and logs? Thanks
0
votes
0answers
38 views

Pre-calc complex number geometry

The equation of the line joining the complex numbers $-5 + 4i$ and $7 + 2i$ can be expressed in the form $az + b \overline{z} = 38$ for some complex numbers a and b. Find the product $ab$. Maybe ...
0
votes
2answers
61 views

Prove symmetry of natural logarithm

Prove that $f(x)=\ln\sqrt{x^2+1}$ is symmetrical in $x=0$. $\ln\sqrt{(x-a)^2+1}=\ln\sqrt{(x+a)^2+1}$ $\sqrt{(x-a)^2+1}=\sqrt{(x+a)^2+1}$ $(x-a)^2+1=(x+a)^2+1$ $x^2-2ax+a^2+1=x^2+2ax+a^2+1$ ...
0
votes
1answer
57 views

$\frac{x}{y} \ge \frac{a_1}{b_1} \ge \frac{a_2}{b_2}$ and $b_1 \le b_2 \implies \frac{x+a_1}{y + b_1} \ge \frac{x+a_2}{y + b_2}$?

Given $\frac{x}{y} \ge \frac{a_1}{b_1} \ge \frac{a_2}{b_2}$, where $x,y,a_i,b_i$ are positive numbers. I would like to prove the following: Claim: If $b_1 \le b_2$, then $\frac{x+a_1}{y + b_1} ...
0
votes
1answer
41 views

Draw graph of $y=x^2$ using these equations

Draw graph of $y=x^2$ using these equations: $x^2-4x+3=0$ $x^2-7=0$ $x^2-2x+5=0$ I don't understand how to put these equations in a form from which they are ready to be plotted on a graph. Do I ...
0
votes
3answers
49 views

calculate the derivative of $x + 1/x$ using the definition?

Calculate the derivative of $x + 1/x$ directly from the definition of the derivative $$ \lim_{h\to0}\frac{f(x+h)-f(x)}{h} $$ I think this is the first step: $((x+h) + 1/(x+h) -(x+1/x) )/h$? but I'm ...
0
votes
1answer
46 views

On the existence/applications of infinitely-nested functions

Inside a previous question, one particular nested function shown is the known tetration. This "kind" of arbitrary repeated functions has always intrigued me, because inside their properties lie so ...
0
votes
1answer
36 views

Precalculus word problem [closed]

Please help me solve this problem: A warehouse is being designed. The brick walls will cost $\$3$ per square foot to build. The roof and floor of the warehouse are flat and square and will cost $\$24$ ...
2
votes
1answer
58 views

procedure of proving that a number is rational [closed]

How can I prove the following ? :
0
votes
2answers
33 views

Help with Evaluating a Function

Find the difference quotient $$\frac { f(a+h)-f(a) }{ h } ,\quad where\quad h\neq 0$$ for the function: $$f(x)=\frac { x }{ x+1 }$$ Steps that I took to try and solve this: $$\frac { \frac { a+h }{ ...
0
votes
2answers
36 views

Solve $yx^2-zx+v=0$ for x?

I am having trouble solving $yx^2-zx+v=0$. $y$,$z$, and $v$ are constants. I cannot just plug into the quadratic formula whilst solving this. I'll post what I have done even though I do not think I am ...
1
vote
4answers
153 views

Factoring the following polynomials

Factorize the following polynomial: $t^3 -9t +8$ $t^6 -91t^2 +90$
2
votes
3answers
257 views

Simplifying the sum of powers of the golden ratio

I seem to have forgotten some fundamental algebra. I know that: $(\frac{1+\sqrt{5}}{2})^{k-2} + (\frac{1+\sqrt{5}}{2})^{k-1} = (\frac{1+\sqrt{5}}{2})^{k}$ But I don't remember how to show it ...
0
votes
1answer
45 views

Every non-empty subset of the integers which is bounded above has a largest element.

I was reading a proof about every non-empty subset of the integers which is bounded above has a largest element, but i have troubles in one step. Here is the proof: Since $S$ is a non-empty subset of ...
1
vote
1answer
17 views

Rate of change vs. average rate of change of a function

What is the difference between the both? I am trying to find a separate definitions for both in a textbook, but it only defines average rate of change. I know that average rate of change is: $$\frac ...
0
votes
2answers
22 views

Find angle $\alpha$ from a complex vector

I'm trying to solve this problem from a Russian book: Find the angle which is needed to rotate the vector $3\sqrt{2} + i2\sqrt{2}$ to obtain the vector $-5+i$. EDIT: $\tan\dfrac{\pi}{6} \neq ...
4
votes
6answers
105 views

When $n$ is divided by $14$, the remainder is $10$. What is the remainder when $n$ is divided by $7$?

I need to explain this to someone who hasn't taken a math course for 5 years. She is good with her algebra. This was my attempt: Here's how this question works. To motivate what I'll be doing, ...
2
votes
2answers
56 views

Calculate $n$ points having equal cartesian distance over a single sine wave

I'd like some help figuring out how to calculate $n$ points of the form $(x,\sin(x))$ for $x\in[0,2\pi)$, such that the Cartesian distance between them (the distance between each pair of points if you ...
3
votes
2answers
113 views

Simplify rational expression

How do I simplfy this expression? $$\dfrac{\frac{x}{2}+\frac{y}{3}}{6x+4y}$$ I tried to use the following rule $\dfrac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot \frac{d}{c}$ But I did not get the ...
0
votes
0answers
28 views

Algebraic solution to this problem?

The problem statement: The reciprocal of $x$’s non-integer decimal part equals $x + 1$, and $x > 0$ Then I'm asked to compare $x$ with the value $\sqrt2$ I can find by picking values for $x$, ...
-1
votes
2answers
53 views

Proof of period of $f(ax+b)$

I have been taught that $f(x)$ is called a periodic function with period $T$ if $$f(x)=f(x+T)$$ This I understand completely. Also I have been taught that $$f(ax)=f(ax+T/|a|)$$ if $f(x)$ has a period ...