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
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1answer
87 views

Sum of $4$ numbers equal to $180$

The sum of $4$ numbers equal to $180$ such that the first number over the second number equal to the fourth number over the third number .How to find these numbers ?
1
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3answers
113 views

Problem with algebra homework

I don't know how to solve this question, can anyone help? $$x^2-2x+1=0$$ How do I solve for $x$? I'm confused. This is for Algebra 1, homework. I don't understand how teacher said use ...
2
votes
2answers
68 views

Area of an elliptic?

I'm looking for an analytic way to calculate the area of an elliptic described by $${x^2 \over a^2} + {y^2 \over b^2}=c^2$$ I saw it before, but now i've forgotten. I remember we set $x=a \cos x$ and ...
11
votes
2answers
260 views

Prove this inequality $a^{\frac{a}{b}}b^{\frac{b}{c}}c\geq1$

Please help me to prove this inequality. Assume $a,b,c>0$ and $abc\geq1$ then $a^{\frac{a}{b}}b^{\frac{b}{c}}c\geq1$. Thanks.
4
votes
1answer
271 views

How to find the smallest positive integer $K$ such that $(K -\lfloor\frac{K}{2}\rfloor + 1)(\lfloor\frac{K}{2}\rfloor + 1) \geq N$

I am writing a program and I would need an explicit formula for the following: The smallest positive integer $K$ such that: $$\left(K - \left\lfloor\frac{K}{2}\right\rfloor + ...
26
votes
5answers
1k views
4
votes
5answers
344 views

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$

Prove:$|x-1|+|x-2|+|x-3|+\cdots+|x-n|\geq n-1$ example1: $|x-1|+|x-2|\geq 1$ my solution:(substitution) $x-1=t,x-2=t-1,|t|+|t-1|\geq 1,|t-1|\geq 1-|t|,$ square, $t^2-2t+1\geq ...
3
votes
7answers
458 views

prove:$\left|x+\frac{1}{x}\right|\geq 2$ [duplicate]

prove: $\left|x+\frac{1}{x}\right|\geq 2$ Can I just use $\left|\left(\sqrt{x}\right)^2+\left(\frac{1}{\sqrt{x}}\right)^2\right|\geq 2$ and ...
2
votes
2answers
59 views

Re-arranging the equation $d=v_i\,t+\frac{a\,t^2}{2}$ to get $t$ equation?

I'm trying to find $t$ equation by re-arranging the equation $\left(d=v_i\,t+\frac{a\,t^2}{2}\right)$ but I'm facing problem because the variable $t$ is existed in two terms. My try (uncompleted ...
0
votes
2answers
57 views

Can I change the order of two terms when factoring: $x^2(x^2-4-3x)$ to $x^2(x^2-3x-4)$?

I'm doing homework and I'm stuck on this assignment: $$x^4 - 4x^2 - 3x^3$$ I figured this would equal $$x^2(x^2-4-3x)$$ Now I know if I would change the order to $$x^2(x^2-3x-4)$$ I can factorise ...
0
votes
2answers
62 views

Having Just one root

For which values of $k$ the equation $3x^3+5x\sqrt{x}+k^2-1=0$ has just one root? $k>\sqrt{2}$ $k\neq 1,-1$ $|k|>1$ $|k|<1$ I think 4 is the right choice.
1
vote
4answers
474 views

Prove that $\cot(A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B}$

The question is: Prove that: $$ \cot(A+B)=\frac{\cot A\cot B-1}{\cot A+\cot B} $$ I have tried expanding it as $\dfrac{\cos(A+B)}{\sin(A+B)}$ and $\dfrac{1}{\tan(A+B)}$.
2
votes
2answers
91 views

Solving Bessel integration

What would be the solution of the bessels equation, $$b=k A(t)\int_0^{\infty} J_0 (k \rho) e^ \frac{-\rho^2}{R^2} \rho d \rho$$ Can I sove that by using this formulation? $$c= \int_0^{\infty}j_0(t) ...
0
votes
1answer
54 views

Elementary prealgerbra

Problem 1 Given to find sum of 1+2+3+4+5+6+7+8+9 I can solve as 1+9+2+8+3+7+4+6+5 45 My question is how can I use associate or commutative property to justify the reordering as above to find the ...
1
vote
3answers
225 views

Proof by Cases: $\operatorname{max}\{x,y\} + \operatorname{min}\{x,y\}=x+y$

So I'm told to "[u]se proof by cases to prove that $\operatorname{max}\{x,y\} + \operatorname{min}\{x,y\}=x+y$ for all real numbers $x$ and $y$." What does this mean?
0
votes
2answers
67 views

Changing from x to y $x = y(4-y)$

$$x = y(4-y)$$ I am guessing I need some pretty advanced math to solve this for y. I am trying to use the shell method and I have to use opposite terms of the rotation axis so I am rotating around y ...
1
vote
2answers
605 views

Find the derivative of $y = f(x^2 - 2x + 7)$ where $f'(10) = 2$

Determine the derivative if $y = f(x^2 - 2x + 7)$ and $f'(10) = 2$ Ok so honestly, I dont know how to solve this, or even know where to start. All i know is that we are given a point $(10, 2)$. But ...
0
votes
0answers
106 views

Simplify the math expression $M=(1-b)^{N}+(N-4)(1-b)^{N-1} b+ \sum^{N}_{i=1} 2^{2i-2} \Delta^2 (1-b)^{N-1}b-(1-2b)^2$

Can someone help me to further simplify the following expression? Here, $0<b<1$ and we can assume that $b$ is small. $\Delta$ is a constant. Thank you $M=(1-b)^{N}+(N-4)(1-b)^{N-1} b+ ...
4
votes
3answers
149 views

Why $\sum_{k=1}^n \frac{1}{2k+1}$ is not an integer?

Let $S=\sum_{k=1}^n \frac{1}{2k+1}$, how can we prove with elementary math reasoning that $S$ is not an integer? Can somebody help?
0
votes
2answers
204 views

Mix-problem with percentage

A can is containing coffee and another can is containing exactly the same amount of milk. We take a spoon of coffee and mix it in the milk can. Then we take a spoon from the mix we obtained and mix it ...
-1
votes
1answer
60 views

State the equation of the line with $x$-intercept $x=5$ and $y$-intercept $y=-2$

I really don't know how to even state the equation. I only know how to find out the $x$ and $y$ intercepts when I know the equation of the line. However, the options are: a. $ 2x - 5y + 10 = 0$ b. ...
1
vote
2answers
126 views

Determining an expression problem

Determine an expression, in simplified form, for the slope of the secant $PQ$ with $P(1,2)$ and $Q(1+h, f(1+h))$ where $f(x) = 2x^2$ I don't even know how to approach such a question. Any help ...
-1
votes
2answers
102 views

Which of the following is not used to determine the slope of a function algebraically?

I dont know the answer to the above question. I think it is the slope of a secant line but I'm not sure ...
0
votes
1answer
106 views

Norm inequality (supper bound)

Do you think this inequality is correct? I try to prove it, but I cannot. Please hep me. Assume that $\|X\| < \|Y\|$, where $\|X\|, \|Y\|\in (0,1)$ and $\|Z\| \gg \|X\|,\|Z\| \gg \|Y||$. prove that ...
3
votes
3answers
693 views

The Product Rule of Square Roots with Negative Numbers

In the statement $\forall a, b \geq0, \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, why is it necessary to restrict $a$ and $b$ to being $\geq 0$? It seems that one should be able to say, for example, ...
4
votes
3answers
175 views

Why don't we define division by zero as an arbritrary constant such as $j$? [duplicate]

Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is ...
1
vote
1answer
112 views

Trigonometric manipulation of complex number, how does this step occur?

I was reading the section about DeMoivre, and my book showed how to derive his formulas. The next part is supposed to be about finding roots of complex and real numbers. Roughly, it says: "Let $z$ be ...
0
votes
1answer
58 views

Find the integer solutions

What are the pairs $(A,N)$ where $A,N$ are integers such that the following equation is satisfied: $\large A=\frac{-6+\sqrt{144-12N^2}}{6}$ I know that we should have: $k^2=144-12N^2$ for some ...
3
votes
1answer
421 views

Find $\cos(2\alpha)$ given $\cos(\theta -\alpha)$ and $\sin(\theta +\alpha)$

My question is: If $\cos(\theta -\alpha) = \frac{3}{5}$ and $\sin(\theta +\alpha) =\frac{12}{13}$, find $\cos(2\alpha)$. Attempt I: \begin{align*} &\cos^2(\theta -\alpha)+\sin^2(\theta ...
0
votes
1answer
133 views

Optimizing $x^2+y^2$ from two given equations? [duplicate]

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to: $$2x^2+5xy+3y^2=2$$ and $$6x^2+8xy+4y^2=3$$ Note: Calculus is not allowed. I tried everything I could but whenever I got for ...
4
votes
2answers
294 views

$4$ women and $2$ men are being interviewed. Find the probability the women will be interviewed first.

My Calculations: $$\frac{4}{6}\times\frac{3}{5}\times\frac{2}{4}\times\frac{1}{3} = \frac 1 {15}$$ Is that correct?
3
votes
1answer
65 views

Given that $a^2(a+k)=b^2(b+k)=c^2(c+k)$, find the value of $1/a+1/b+1/c$

Given $$a^2(a+k)=b^2(b+k)=c^2(c+k)$$ find the value of $1/a+1/b+1/c$. I tried to derive a relation from the equality but it did not help my cause.
5
votes
2answers
161 views

How many times can a $4^{th}$ degree polynomial be equal to a prime number?

If $f(x)$ is a $4^{th}$ degree polynomial with integer coefficients, what is the largest set ${x_1, x_2, x_3, ...x_n}$ (where $x_i$ are integers) for which $|f(x_i)|$ is a prime number? Things I ...
2
votes
1answer
37 views

Stuck on rearranging of this equation

I need to get from $[(1-p)f+p(1-f)](1+v)-[(1-p)(1-f)+pf] = x$ to $(2+v)(f+p-2pf)-1 = x$ but I'm stuck. I'd appreciate any tips on what I should I do after the following. $(f+p-2pf)(1+v) + (f + p ...
-2
votes
1answer
89 views

$\cos x$, $\cos \pi x$, $|\cos \pi x|$ [closed]

Why is the graph of $\;\cos\pi x\;$ like a linear function? And $\left|\;\cos\pi x\;\right|$ like an absolute value function? I know about the absolute function, but isn't it the graph of $\cos x$ ...
1
vote
1answer
45 views

Smallest value of n for two algoritms with a certain running time

If one algorithm has a running time of $100n^2$ and another of $2^n$; how can I find the smallest value of $n$ such that the former is faster than the latter? I could do: $100n^2 < 2^n$ then ...
0
votes
1answer
58 views

Extract time frames from days

I am a computer programmer, and I like to performe some maths and I am not sure for the correct method to use. More specific, I am creating an application that charge a client based on time usage of ...
1
vote
3answers
601 views

Solve the following equation: $x^4- 2x^2 +8x-3=0$

Solve the following equation: $$x^4- 2x^2 +8x-3=0$$ We get 4 equations with 4 variables. But that is too difficult to solve. My try: Let $a,b,c,d$ be the roots of the equation. $$a+b+c+d=0$$ ...
1
vote
2answers
42 views

How to find the maxima?

This is a simple question : Find the maximum value of $\frac { 1 }{ { x }^{ 2 }-6x+2 }$ I rewrote ${ x }^{ 2 }-6x+2$ as $(x-3)^{2} - 7$, now when this is min, the original function is max, thus the ...
6
votes
4answers
732 views

If $a+\sqrt{b}=c+\sqrt{d}$ does $a=c$ and $b=d$?

If $a+\sqrt{b}=c+\sqrt{d}$ does $a=c$ and $b=d$? I am grading some problems and I don't think this true, but all of a sudden I am doubting myself...
0
votes
2answers
41 views

Simple algebra loss calculation

Kylie bought an item for $x$ and sold it for \$10.56. If Kylie incurred a loss of $x$ percent, find $x$. The answer is apparently "12 or 88" but I cannot see how they got there. I have tried ...
2
votes
2answers
136 views

Prove an inequality concerning $\sqrt[3]{4a^3+4b^3}+\sqrt[3]{4b^3+4c^3}+\sqrt[3]{4c^3+4a^3}$

Let $a,b,c$ be positive. I need to prove $\sqrt[3]{4a^3+4b^3}+\sqrt[3]{4b^3+4c^3}+\sqrt[3]{4c^3+4a^3}\leq \dfrac{4a^2}{a+b}+\dfrac{4b^2}{b+c}+\dfrac{4c^2}{c+a}$ Thanks!
0
votes
1answer
49 views

Searching for the ratio in alloys

Two alloys A and B are composed of two basic elements. The ratios of two compositions of two basic elements in the two alloys are 4:3, 5:4 respectively. A new alloy X is formed by mixing the two ...
0
votes
2answers
55 views

How to determine solutions: $\;2^x=3^y=36^{-z}\; \implies \frac1x +\frac 1y +\frac 1{2z} = \quad ?$

If $$2^x=3^y=36^{-z}\;$$ then $$\frac1x +\frac 1y +\frac 1{2z}$$ is equal to a) $\;0$ b) $\;1$ c) $\;-1$ d) none of theses How to solve this problemplease explain it
0
votes
1answer
70 views

Pythagorean motion

At the same moment two particles start respectively from vertices $B$ and $C$ of triangle $ABC$ which has a right angle at $C$. The particles move at constant speeds and arrive at vertex $A$ at the ...
1
vote
1answer
112 views

Re-arranging the equation $L=\sqrt{a^2\sin^2t+b^2\cos^2t\,}$ to find $\left(t\right)$?

How can I re-array the equation $L=\sqrt{a^2\sin^2t+b^2\cos^2t\,}$ to find the equation of $\left(t\right)$ ? $t=\,?$ I tried to solve it but I'm stuck at: $L^2=a^2\sin^2t+b^2\cos^2t$ ...
0
votes
1answer
99 views

Re-arranging the equation $t=\arctan\left(\frac{a}{b}\tan\theta\right)$ to find $\theta$

How can I re-array the equation $t=\arctan\left(\frac{a}{b}\tan \theta\right)$ to find the equation of $\theta$. $\theta=\,?$ Actually I tried this equation: ...
0
votes
1answer
57 views

Is it possible to find the term (variable) $c$ from the equation $a^2=\sqrt{b^2+c^2}$

If I have the equation $a^2=\sqrt{b^2+c^2}$. Is it possible to me to find the term (variable) $c$ from it ? $c=\,?$
0
votes
1answer
56 views

Any way to tell when an algebraic expression takes on values that are a square?

Say I have the expression $256x^2 -480x$. As a polynomial this isn't a perfect square. However that doesn't stop it from taking real values that are a perfect square for given x, such as x = 8. Is ...
1
vote
2answers
77 views

If $A=(-4,0)$ and $B=(4,0)$, what is the locus of points $P$ such that $|AP-BP|=16$? Does it even exist?

I am stuck in this question for about a week: If there are points $A$ and $B$ such that $A(-4,0)$ and $B(4,0)$ then what is the locus of points $P$ such that $|AP-BP|=16$? I think this is a ...