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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11 views

Let $P$ be a 4-th degree real polynomial with 5 conditions given. How to compute $P(4)$?

Yesterday I was math tutoring a 18-years old girl. And she asked me for the following problem: given $P\in\Bbb R[X]_4$, i.e. $P$ a real polynomial of degree exactly $4$, such that: $P(1)=0$ It has a ...
0
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2answers
29 views

Show that $f(x)=2x^2+4x+5$ is positive for all real values of $x$.

Show that the function $f(x)=2x^2+4x+5$ is positive for all real values of $x$. And find its minimum value. Hence show that $0<\frac{6}{f(x)}\leq2$.
2
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4answers
46 views

Easiest way to solve the equation $x^\frac 43 = \frac {16}{81}$

What would be the easiest way to solve this? $$x^\frac 43 = \frac {16}{81}$$ I saw this in class and have no clue how did they get $$x = \frac 8{27}$$
2
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3answers
26 views

A milkman has $80\%$ of milk in his stock of $800$ litres of adulterated milk. How much $100\%$ milk to be added to give certain purity?

Problem: A milkman has $80\%$ of milk in his stock of $800$ litres of adulterated milk. How much $100\%$ milk to be added to it so that the purity of milk is between $90\%$ and $95\%$ Let $x$ litres ...
0
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2answers
29 views

Deduce inequality between geometric mean and power mean from AM-GM

Below is exercise 20 from Tom Apostol's "Calculus" Vol. 1 (2nd edition). I need help solving part (b). The geometric mean $G$ of $n$ positive real numbers $x_1, x_2, ..., x_n$ is defined by the ...
0
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1answer
9 views

Computing shortest path including specific edge

Consider the weighted undirected graph with $4$ vertices, where the weight of edge $\{i, j\}$ is given by the entry $W_{i, j}$ in the matrix $W$. $$W = \begin{bmatrix} 0&2&8&5\\ ...
1
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1answer
26 views

The maximum possible size of $R$ is_____?

A function $f : N^+ → N^+$, defined on the set of positive integers $N^+$, satisfies the following properties: $f(n) = f(n/2)$ if $n$ is even $f(n) = f(n+5)$ if $n$ is odd Let $R = \{i|∃ j : f(j) = ...
1
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1answer
43 views

Explain the last line of this proof (it's an identity I can't figure out)?

I can't understand the equality in equation 4 in this paper. I understand everything up until that point. I see how $$j \binom{n-p}{j}=(n-s)\binom{n-s-1}{j-1}$$ but I'm not sure how they used that ...
0
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0answers
18 views

True of False inequality graphing questions (plug in )

Point $(6,y)$ is a solution of the inequality $12y+x>0$ for any value of $y$. I got false since $y$ could be negative $100$ and that plus $6$ would be less than $0$. Is that correct? Also, in ...
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5answers
313 views

Is there a way to prove this exponential inequality?

I came across this proposition while trying to prove that a function was injective: if $a>b$ then $a^a>b^b$, where $a$ and $b$ are real numbers bigger than 1 . Intuitively it (somehow) makes ...
0
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5answers
75 views

Solve $(x-5)/(x+1)= (x-5)/(x+3)$.

Someone is asked to solve the following equation: $(x-5)/(x+1)= (x-5)/(x+3)$. This person respond "There is no solution. Cross multiply to get $(x-5)(x+3)=(x-5)(x+1)$. Divide both sides by $x-5$ and I ...
0
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1answer
22 views

How to find the quadratic equation with 2 variables, given 2 functions independent of the other variable.

I've been asked to find the function for f(a,b). given that f(a,0) = .02*((a-2)^2) + 170 f(0,b) = 0.00125 * (b^2) + 170.08 Can you give me a suggestion of how I ...
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4answers
76 views

How to prove $ 1+1 + 2+ 2^2 +2^3 + \cdots + 2^{30}= 2^{31}$? [on hold]

This is true? $ 1+1 + 2+ 2^2 +2^3 + \cdots + 2^{30}= 2^{31}$, How to prove?
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1answer
52 views

Finding square of cube? [on hold]

A cube is built using $64$ cubic blocks of side one unit. After it is built, one cubic block is removed from every corner of the cube. The resulting surface area of the body (in square units) after ...
0
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4answers
84 views

Factoring $x^4 - 16$

I was following a calculus tutorial that factored the equation $x^4-16$ into $(x^2 +4) (x+2)(x-2)$. Why is the factorization of $x^4-16 = (x^2 + 4)(x+2)(x-2)$ rather than $(x^2 - 4)(x^2 +4)$?
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1answer
17 views

How to form 3 groups and allocate them different district?

At an election three districts are to be canvassed by $10, 15$ and $20$ men respectively. If $45$ men volunteer, in how many ways can they be allotted to the different districts? Three groups ...
1
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1answer
39 views

Distinct real roots .

Problem : If $|\log(x)| - px = 0$ has three distinct real roots then the range of $p$ will be ? My attempt : I tried to see the problem graphically and made the graph. So I am able to see that ...
1
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2answers
69 views

The Triple Number Game (n + n + n)

I work at the Science Museum on weekends, and I sometimes present the following puzzle: $$0\;0 \; 0 = 6$$ $$1 \; 1 \; 1 = 6$$ $$2 \; 2 \; 2 = 6$$ $$3 \;3 \; 3 = 6$$ $$\vdots$$ $$n\;n\;n=6$$ The idea ...
1
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1answer
26 views

Proving a reduction formula. $\cos^n (2x)$

Establish a reduction formula for $$\int \cos^n (2x)dx$$ My attempt, Let $I_n=\int \cos^n 2x dx$ $=\int \cos^{n-1}2x (\cos 2x dx)$ Let$$u=\cos^{n-1}2x$$ $$du=-2(n-1)\cos^{n-2}2x (\sin 2x)dx$$ ...
3
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2answers
73 views

Fraction Sum Series

This question was asked in (selection) IMO for 8th graders. $1/2 + 1/6 + 1/12+ 1/20 + 1/30 + 1/42 +1/56 + 1/72 + 1/90 + 1/110 +1/132$ I have noticed that it can be written as $1/(1*2) + 1/(2*3) ...
0
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0answers
47 views

Are polynomial roots special? [on hold]

Most functions have roots, and relations can also have roots (including complex roots) as well. They are where $f(x)=0$ for some $x$. (I'm considering only functions that actually will have roots.) ...
1
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4answers
73 views

Is there any value for $x$ that would make the statement $(x+3)^3 = x^3+3^3$ true?

Is there any value for $x$ that would make the statement $(x+3)^3 = x^3+3^3$ true? I understand that when factored out, you have $(x+3)^3 = x^3+9x^2+27x+27$ as opposed to the other side which is ...
1
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1answer
24 views

Precision of Manual Vector Addition

I learned the fundamentals of vectors and basic (e.g. addition, dot product) vector operations in a Trigonometry course, and they're being reintroduced in the Physics I course I just began. My ...
0
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1answer
15 views

Generate a formula to calculate affect on continously compound interest if a constant salary is continously added to principle

I'm feeling stupid with a simple algebra question. I'm hoping someone can help me figure out why something I know should be obvious just isn't clicking for me. I won't go into actual details of the ...
1
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3answers
74 views

If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$.

If $a$ is a real root of $x^5 − x^3 + x − 2 = 0$, show that $\lfloor a^6 \rfloor = 3$. Obviously since this is a 5th degree polynomial, solving it is not going to be possible (or may be hard). ...
1
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3answers
40 views

The line $x+\sqrt{3} y-10=0$ makes an angle of $150$° with the positive sense of the $x$-axis. How can this be proven?

I cant figure out how this is correct. I know that $\tan(a)=m$ of a line but I cant figure this out. Could someone show how to prove the line makes an angle of $150$° with the positive $x$-axis? I ...
1
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2answers
42 views

Solve $x + \frac{ 1 }{y+1/3}=38/3$ in the set of natural numbers

The following equation should have a solution with $x,y$ being natural numbers. I cannot find it. Is there such solution? $$x + \frac{ 1 }{y+1/3}=\frac{38}{3}$$
0
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4answers
22 views

Evaluating simple summation

can someone help with this summation. Seems simple, but... I have tried several options but cannot see the rule. $\displaystyle 1-a+a^2-a^3+...a^{2008}-a^{2009}+\frac{a^{2010}}{1+a} {\text{ when}}\ ...
1
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0answers
31 views

reduction formula for $\int \tan^n (2x)dx$

Establish a reduction formula for $$\int \tan^n (2x)dx$$ My attempt, Let $I_{n}=\int \tan^n (2x)dx$ $=\int \tan^2 (2x) \tan^{n-2} (2x)dx$ $=\int (\sec^2 (2x)-1)\tan^{n-2}(2x)dx$ $=\int ...
0
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1answer
31 views

how to scale a value from 0-1 to 0-5

I have these facts: the user (X) rates the item (I) by (4/5) the item (II) is 0.5 similar to item (I), 0.5 means 50% (the scale is from 0 to 1) then I can say (according to my business model) that ...
0
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3answers
15 views

How can I find $y$ coordinate of a straight line at a specific $x$ value

Lets say I have a straight line between $p_1=(-2, -0.5)$ and $ p_2=(0.25, 0.5)$. How can I find the value of $y$ when $x=-1$? I have tried to solve this the whole day without finding an answer, I ...
2
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2answers
57 views

How to solve without involving hyperbolic function.

How to solve this integral without involving hyperbolic functions? $$\int \frac{1}{4-5\sin^2 x}dx$$ The answer is $\frac{1}{4}(\ln (\sin x+2 \cos x)-\ln(2\cos x-\sin x))+c$
3
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4answers
75 views

Show that $8x^4 −16x^3 +16x^2 −8x+k = 0$ has at least one non-real root for all real $k$. Find the sum of the non-real roots

Show that $8x^4 −16x^3 +16x^2 −8x+k = 0$ has at least one non-real root for all real $k$. Find the sum of the non-real roots. Since this polynomial looks so symmetric, I think factoring it might ...
6
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1answer
65 views

Basic question $|x^2| < 9$

I have a rather basic question. Let's assume that $|x^2| < 9$, where $x\in \mathbb{R}$. Then everyone knows that $x \in$ (-3,3). However, I have trouble arriving at the answer based on basic ...
0
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2answers
64 views

Integer value of the given radical: $\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}}$ [on hold]

What is the value of $$\sqrt{2+\sqrt{5}-\sqrt{6-3\sqrt{5}+\sqrt{14-6\sqrt{5}}}}$$ I don't know how to simplify it?
1
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2answers
53 views

Where did my simplification go wrong? Sum and difference formula simplification

I'm struggling with the following: We are to use the sum and difference formulas to find the exact value of the expression. The problem is simplification has been tough. As a last resort I decided to ...
0
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1answer
23 views

Is there a unique solution to this upstream/downstream canoe rowing proposition?

A man jumped into his canoe and paddled upstream for one mile. After this, he continued for another fifteen minutes. Having arrived at his destination, he then turned around and paddled downstream, ...
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5answers
45 views

How to write absolute value as a “true” function

Here's the basic absolute value ... what? \begin{align*} |x| = \left\{ \begin{array}{r@{\quad \mathrm{if} \quad}l} x & x > 0, \\ 0 & x = 0, \\ \!\! -x & x < 0. \end{array} ...
0
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2answers
26 views

Factor Theorem (Finding values of a and b)

Question: The polynomial $p(x) = 2x^3 - ax^2 + bx + 48$ has $(x-4)$ as a repeated factor, find the values of $a$ and $b$. What I have attempted if $x-4$ is a factor then $x = 4$ is a ...
0
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2answers
40 views

solving system of equations involving imaginary numbers

What are the values of $a,b,c$ given the system of equations given below: $a+b+ab=i$ $b+c+bc=2i$ $c+a+ac=3i$
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3answers
113 views

What is my special quadratic?

Start with $f(x)=x^2+bx+c$. Then, attempt to solve for $x$ in $f(x)=x$. It is easily found that $x=\frac{1-b\pm\sqrt{(b-1)^2-4c}}{2}$. Then, start again with $f(x)=x$ and apply the function $f$ to ...
3
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0answers
20 views

Is this a valid way for performing polynomial division?

While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ...
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0answers
29 views

How do I get these values? [on hold]

I want to know how to get these answers: $A(12836.3)=42227.7$ $A=3.28971$ (phase) $\theta+46.7364=0$ $\theta=-46.7364$ from this equation. I am confuse is there a formula or a trig. identity? ...
4
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0answers
36 views

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$).

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$). My effort Rearranging our equation we have : \begin{array}{c} 1990[x]+1989[-x]&=1 \\ ...
1
vote
1answer
30 views

Struggling with BIDMAS.

This question came up an I'm not sure about it: You need to simplify leaving the answer in standard form: $\dfrac{2((3-3^2)^2)}{3+\sqrt{4^2-7}}$ I struggle to work it out myself. When I used a ...
6
votes
4answers
111 views

Solve equation $\frac{1}{x}+\frac{1}{y}=\frac{2}{101}$ in naturals

My try was $$\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}=\frac{2}{101}\\x+y=2k,xy=101k\\x=2k-y\\y(2k-y)=101k\\2ky-y^2=101k\\y^2-2ky+101k=0\\y=k+\sqrt{k^2-101k}\\x=k-\sqrt{k^2-101k}$$ Now $\sqrt{k^2-101k}$ ...
2
votes
1answer
62 views

Why do you need absolute value when taking $\sqrt{\cos^2(x)}$

$$\sqrt{\cos^2(x)} = |\cos(x)|$$ Is this on the right track? If you have an underlying $\cos(x)$ that is negative, and then you square it, you will now have $\cos^2{x}$, which is positive. But, if ...
4
votes
2answers
48 views

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$? Because of the nature of the square root function, its derivative monotonically decreases. so ...
0
votes
2answers
37 views

What is my mistake

Spot my mistake: $$\frac{\left(\text{P}_1+\text{P}_2+\dots+\text{P}_n\right)-\left(\text{Z}_1+\text{Z}_2+\dots+\text{Z}_n\right)}{n-m}\le-\ln(50)$$ ...
0
votes
2answers
25 views

Simplifying Expression Factorial Expression

I'm confused as how I'm meant to simplify this:$$\frac{(n-2)!}{(n-2-r)!}$$ I have other factorial questions where the variable isn't present in the top factorial like the question above and I'm ...