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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1
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2answers
23 views

Finding the points where a circle intersects an axis

A circle has the equation: x²+y²+4x-2y-11 = 0 What would be the coordinates of the points where the circle intersects with the y-axis and how would you calculate it?
-2
votes
2answers
48 views

How to simplify $(x+4)^2$? [closed]

I have been getting this problem in various situations. But I always get a different answer.
0
votes
3answers
56 views

How to solve this word problem on the topic of quadratic equations?

The maths teacher of Mumbai is transferred to another school. The students of Class 10 decided to buy a book for 360 rupees(currency) as a gift for her. On the farewell day, 4 students did not turn up ...
4
votes
1answer
52 views

Why is this answer wrong? (rational expressions)

Simplify the following expression: $$q=\frac { z+9 }{ 5 } +10$$ this is what I got: $$q=\frac { z+9 }{ 5 } +\frac { 50 }{ 5 } $$ $$ q=\frac { z+9+50 }{ 5 } $$ This answer is wrong: $$q=\frac { ...
3
votes
4answers
85 views

If $a,b,c$ are positive, then $(a+b+c)(1/a+1/b+1/c)\ge 9$

The question asks to prove that if "$x_1,x_2,x_3$ are positive numbers show that: $$(x_1+x_2+x_3) \left(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3} \right)\ge 9$$ I've tried to use the fact that the ...
0
votes
0answers
34 views

Need help with excel spreadsheet! [closed]

So I am currently doing an assignment in which I have bought a house using a home-loan. For this part of the question I need to calculate how long it will take to repay the loan. So I have constructed ...
0
votes
2answers
49 views

how shall i find the $n$-th term of this,

How shall I find the $n$-th term of this: $\sqrt{1+2}$ $\sqrt[3]{1+2+3}$ $\sqrt[4]{1+2+3+4}$ $\sqrt[5]{1+2+3+4+5}$ $\sqrt[6]{1+2+3+4+5+6}$ $\sqrt[7]{1+2+3+4+5+6+7}$ all the way to ...
2
votes
0answers
57 views

$\sum_{k=1}^n \lfloor kx \rfloor =$ ?

Let $x$ be a positive real number and $n$ a positive integer , then how may we evaluate $\sum_{k=1}^n \lfloor kx \rfloor $ ? If a closed form doesn't exist then can we at least find an asymptotic ...
1
vote
4answers
58 views

If $x^n=y^n$ and $n$ is odd then $x=y$

Here, we suppose that $x,y\in\mathbb{R}$ and that $x^n=y^n$, where $n$ is odd. I want to prove that $x=y$. Maybe we can use that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$ So, it ...
1
vote
1answer
28 views

How do i solve this to find PMT?

I know this may seem like a stupid question but i've been up late working on this math assignment and this question just isn't working when i transpose it. So this is the formula to find Present ...
0
votes
0answers
18 views

Boehm's and Jacopini's insight and Math

My precalc teacher mentioned "If, then, else" in class today. So from the little programming experience I have, I picked up that the same thing exists in CS. I was wondering what that is called in the ...
0
votes
4answers
24 views

Progressions and ratio problem

The ratio of arithmetic mean and geometric mean of two numbers is 5:4. If the difference between their geometric mean and harmonic mean is (-0.8) find the numbers. I tried using ratio property but I ...
0
votes
4answers
39 views

Time it will take to complete the project if two people work together (word problem)

If Iris spends 3 days and Olivia spends 5 days on a project, 1/2 of the work can be done. If instead Iris spends 5 days and Olivia spend 3 days, then 1/3 of the work is done. How long does it take to ...
-4
votes
1answer
35 views

If a and b are the two solutions to x^2 - x - 2=0, then a+b=? [closed]

If a and b are the two solutions to x^2 - x - 2=0, then a+b= ? A. -1 B. 0 C. 1 D. 3 E. 5
2
votes
0answers
32 views

What is the (currently) optimal root finding algorithm for multivariate functions? [closed]

Let's say we wish to find the roots of the function: $f(x,y,\cdots) = 0 \;,$ so, for a minimal example: $xy - 1 = 0 \; .$ I know there are different methods to solve this problem for the ...
0
votes
1answer
17 views

$P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ on the line that bisects $\angle F_1PF_2$. Prove $|PF_1-PF_2|>|QF_1-QF_2|$.

$\require{cancel}$ Sorry for the grammatical mistake in the title; it was needed to keep the title under 150 characters. $P$ is a point on a hyperbola whose focal points are $F_1$ and $F_2$. $Q$ is ...
0
votes
2answers
38 views

Largest number of pairs that can be added while keeping the population at least 60% male

I'm doing problems from the AoPS Algebra Beginner's book. There's this problem that states the following, At her ranch, Georgia starts an animal shelter to save dogs. After the first three days, she ...
1
vote
2answers
102 views

Can one use logarithms to solve the equations $2=3^x + x$ and $2=3^x x$?

Could someone explain how would you solve: $$2=3^x + x$$ and $$2=3^x \cdot x$$ I can only solve halfway through. And why is $$10^{\log (x)}= x$$ Thanks
1
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2answers
66 views

How to find the domain and range of $f(x) = \sqrt{x^2-2x+5}$?

This is the function: $$f(x) = \sqrt{x^2-2x+5}$$ Edit: normally what I would do is this: Since it's a square root function, the thing inside the root has to be $\ge 0$. So, $(x^2 - 2x+5)\ge 0$. Then ...
0
votes
1answer
26 views

Standard form of trig equations

The standard form for any trig equation is y=Asin(B(x-D))+C (I'm just using sine in the equation). For the "D" which is the horizontal translation, if D is added does the graph move left or right, ...
1
vote
1answer
34 views

Volume of a ellipsoidal shape

I was given the following question: My approach so far was to create a parabolic function: y = 25/2 - (25^2)/392 Then I integrate from x = 0 to x = 14 Volume = 2 * pi * integral of y ^ 2 The ...
-1
votes
2answers
25 views

Graph of $\log_2(2-x)$: what is wrong in my transformational approach?

In the graph of $\log_2(2-x)$ can I have the transformational approach of $\log_2 x$ >> $\log_2(-x)$ >> $\log_2(-x+2)$ or $\log_2(2-x)$ but after all this graph comes wrong but with differential ...
0
votes
3answers
51 views

Factor the expression completely.

$$(a^{ 2 }+1)^{ 2 }-7(a^{ 2 }+1)+10$$ So far I got: $$(a^{ 2 }+1)(a^2+1)-7a^{ 2 }+3$$ I feel like I am going about this the wrong way. I need a push in the right direction.
1
vote
2answers
19 views

Graphing the secant function, $y=2\sec 2\theta$

I am asked to graph the following equation $y = 2 \sec {2\theta}$. Since the equation just has a $ 2\theta$ after the secant, is it correct to say that there is no phase shift? If I would start ...
2
votes
1answer
53 views

Is it possible to accurately calculate an irregularly shaped frustum's volume?

I have the following water basin Now imagine this basin is filled with water to the top, is there anyway to accurately calculate the volume of water stored in it using only top and bottom areas A1 ...
0
votes
5answers
53 views

inequality with absolute value?

Solve the given inequality by interpreting it as a statement about distance on the real line: $$|x+1| \gt|x-3|$$ anyone know how to go about this problem?
0
votes
0answers
42 views

Can the inequality $a^3 + b^3 + c^3 \ge a^2b + ac^3 + b^2c$ be derived from arithmetic-geometric means? [duplicate]

The inequality goes as follow: $$a^3 + b^3 + c^3 \ge a^2b + ac^3 + b^2c$$ Where $a,b,$ and $c$ are positive real numbers. Also, can it be solved using am-gm?
1
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3answers
51 views

Simple use of log

I am struggling to see how we can go from the first expression to the second: $$\begin{align} 2\log_3 12 - 4\log_3 6 &= \log_3 \left ( \frac{4^2 \cdot 3^2}{2^4 \cdot 3^4} \right )\\ &= \log_3 ...
1
vote
1answer
33 views

What is the value of the mean of these numbers?

Given that $13^{a+b}=13^{xy}=13^{13}$, what is the mean of $a, b, x$ and $y$? What I tried: the mean is $\frac{a+b+x+y}{4}$. One can infer that $a+b=13$ so that the mean is $\frac{13+x+y}{4}$. I ...
1
vote
4answers
56 views

Converting repeating decimal to fraction

How do I conver $0.297$ to a fraction, if the 2 and 9 are repeating? The non repeating number is in the middle, so I am not sure how to proceed from here. Any help is appreciated
1
vote
1answer
41 views

Formula to find the value after taking the square root of a number $n$ times?

How can I find the value after taking the square root of a number $n$ times? For example: $\sqrt{a}$, $\sqrt{\sqrt{a}}$, $\sqrt{\sqrt{\sqrt{a}}}$, $\sqrt{\sqrt{\sqrt{\sqrt{a}}}}$ and so on.
0
votes
1answer
65 views

Is there any simple analytic method for solving $\sqrt{x}+y=7$ and $x+\sqrt{y}=11$ simultaneously. [duplicate]

I am thinking of a nice and simple analytic method to solve the following equations simultaneously: $$\sqrt x+y=7;\\x+\sqrt y=11.$$ To my suprise I can't. But, I solve the system numerically using ...
-1
votes
1answer
45 views

Logic Question - Minimal Calculations Required [closed]

Not sure were to start. Have fun.
1
vote
1answer
27 views

Basic graphing - plot v = 10i +4

So I have the function $v=10i+4$ where $v$ is the horizontal axis and $i$ is the vertical axis. Please excuse me for such a basic question but I can't work out how to draw this function. I figure if ...
5
votes
5answers
112 views

How $\sqrt{2}=1+\frac{1}{\sqrt{2}+1}$?

I have found it in the chapter about chain fractionals. I am unable to transform it to such state. $$\sqrt{2}=1+\sqrt{2}-1=?=1+\frac{1}{\sqrt{2}+1}$$
0
votes
3answers
72 views

Find x in this equation

Can you please help me find x in this equation? My knowledge on the rules of algebra is honestly limited. I simply cannot isolate the x on one side. $$\frac{(1+x)^3-1}{x}=3.1836$$
0
votes
1answer
27 views

Solution in terms of Lambert $W$ function

Is it possible to solve equation of the following form using Lambert $W$ function. $$(x-a)^2 = b(e^{-cx} - cx + 1).$$ If not, can it be solved using any other special function??
0
votes
1answer
18 views

Algebraic Equation Vexation

I was asked to help my sister with a bit of precalculus homework and completely drew a blank upon encountering this problem. I believe it was asking to "balance the equation, and set the answer to ...
2
votes
9answers
208 views

Why does $(a+b)^2= a^2+b^2 + 2ab$? Why is the $2ab$ there?

When I was doing research on finding the derivative I came across something strange. If $f(x) = x^2$ you find the derivative by going $$\frac{f(x+h)^2-f(x)^2}{h} =\frac{x^2+2xh+h^2-x^2}{h}.$$ Why ...
0
votes
1answer
36 views

What trig. identity would help solve $2 + \cos(2x) = 3\cos(x)$?

I need help with a homework question that has me puzzled. I need to solve the following equation: $$2 + \cos(2x) = 3\cos(x)$$ I don't see a good trig identity to apply. I tried $\cos(2x) = ...
4
votes
3answers
88 views

The Sum of ${11^{th}}$ power of the roots of the equation ${x^5+5x+1=0}$

The Sum of ${11^{th}}$ power of the roots of the equation ${x^5+5x+1=0}$ ${My\; Try::}$ Let ${x=\alpha\;,\beta\;,\gamma\;,\delta\;,\mu}$ be the roots of the equation ${x^5+5x+1=0}$ So ...
0
votes
0answers
32 views

find gcd$( \lfloor x \rfloor, \lfloor x^a \rfloor)$

Is there a way to find gcd$( \lfloor x \rfloor, \lfloor x^a \rfloor)$ assume that $x>1$ and can assume that $a>0$. Also, if close form doesn't exist are there meaningful lower bounds and upper ...
1
vote
1answer
28 views

Circumference of separate circle

So I have been out of Algebra for a while now. I am trying to help my wife prep for an entrance exam and we ran across this in the practice test: ...
0
votes
2answers
40 views

integrating $\ln(ax)$ in an equation.

The derivative $\frac{d}{dx}\ln{(ax)} = \frac{1}{x}$ What follows is that $\int{\frac{d}{dx}\ln{(ax)}} = \int{\frac{1}{x}}$ And so, $\ln{(ax)} + c_1 = \ln{|x|} + c_2$ where $a, c_1, c_2 ...
4
votes
2answers
24 views

what geometric object is represented (in the complex plane) by the solution of an equation?

The solution to the equation: _ z = 2/z can be described as a geometric object, which? anyone know how to go about this problem? thanks in advance ...
2
votes
1answer
81 views

Minimizing the expression $(1+1/x)(1+m/y)$ over positive reals such that $mx+y=1$

Let $x$ and $y$ be positive real numbers such that $mx+y=1$. Find the positive $m$ such that the minimum of: $$\left( 1 + \frac{1}{x} \right)\left( 1 + \frac{m}{y} \right).$$ is $81$. I have ...
0
votes
1answer
45 views

How can I rearrange $Y =\frac{X}{A+BX}$ to solve for $X$?

I know Y, A, and B but how can I solve this for X? $Y =\dfrac{X}{A+BX}$ The $X$ value is the same number if that matters. I used this equation to solve for $A$ but I want to know how I can plug back ...
0
votes
2answers
50 views

Equating functions: does f=g?

$f(x)=\frac{x^2-2x}{x-2}$ $g(x)=x$ Does $f=g$? I said yes but my homework said they aren't equal.
1
vote
3answers
33 views

Is there any way to express $\theta=c$ as some function of $r$?

I recently found this: Desmos Graphing calculator. I tried to plot the equation $\theta=45$ but it gave me an error: Sorry, you can't graph $\theta$ as a function of anything yet. So I started ...
0
votes
1answer
45 views

Not understanding the solution to this rational expression

$$\sqrt { 1+\left(\frac { x }{ \sqrt { 1-{ x }^{ 2 } } } \right)^{ 2 } } $$ I have done the following: $$\sqrt { 1+ \frac { x^2 }{ { 1-{ x }^{ 2 } } } } $$ $$\sqrt{ \frac { 1-x^{ 2 } }{ 1-x^{ 2 ...