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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1answer
17 views

Solve Equation for x and state any restrictions on the variable difficulties

I am a high school student in Algebra II and while I normally have no trouble with problems dealing with algebraic equations, I simply cannot muster the answer to this question. Solve for x. State ...
6
votes
1answer
89 views

How to explain combinatorial identities?

The setup of binomial expansion formula can be traced by two paths, one of which is "pure" proof by induction (using properties of combinatorial numbers), the other is "practical" comprehension by ...
2
votes
2answers
162 views

Constant such that $\max\left(\frac{3}{3-2c},\frac{3a}{3-2d},\frac{3b}{3-2e}\right)\geq k\cdot\frac{2+3a+4b}{9-c-2d-3e}$

What is the greatest constant $k>0$ such that $$\max\left(\frac{3}{3-2c},\frac{3a}{3-2d},\frac{3b}{3-2e}\right)\geq k\cdot\frac{2+3a+4b}{9-c-2d-3e}$$ for any $0\leq b\leq a\leq 1$ and $0\leq c\...
4
votes
3answers
66 views

Prove that $\max\{a_i \mid i \in \{0,1,\ldots,1984\}\} = a_{992}$

Consider the expansion $$\left(1 + x + x^{2} + x^{3} + x^{4}\right)^{496} = a_{0} + a_{1}x + \cdots + a_{1984}\,\,x^{1984} $$ Prove that $\max\left\{a_{i} \mid i \in \left\{0,1,\ldots,1984\right\...
3
votes
3answers
104 views

On the proof that $\left|\frac{z_1-z_2}{1-z_1\bar{z_2}}\right|\lt 1$

Question: Prove that $\left|\dfrac{z_1-z_2}{1-z_1\bar{z_2}}\right|\lt 1$ if $|z_1|\lt1$, $ |z_2|\lt 1$ My solution: I had no idea how to go about this one so instead I started simplifying the ...
0
votes
2answers
66 views

Eliminate $x_1,x_2,y_1,y_2,c$ from the following equations

I need to eliminate $x_1,x_2,y_1,y_2,c$ from the following equations.What would be the correct ( and quick ) technique to do so? $x_1y_1=1$ $y_1=4x_1+c$ $x_2y_2=1$ $y_2=4x_2+c$ ...
1
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1answer
48 views

Substituting cot(x) with an identity in equation

Assume: $$\cos(x) / \cot(x) = 0\tag A$$ I rewrite it as $$\cos(x)/ \left(\cos(x)/\sin(x)\right) = 0\tag B$$ and get $$\sin(x) = 0\tag C$$ This implies $x = 180^\circ\times k$ But, under (B), if $\sin(...
2
votes
3answers
59 views

Writing domains: $∈$ or $⊆$?

Usually when we write domains for functions (e.g. $f(x)=x^2$) in set notation, we would write something like this: $$D=\{x∈ℝ\}$$ This means that all values of x are part of the set of real numbers. ...
2
votes
1answer
73 views

How to get a whole number from $y = \frac{1}{x + 2}$

How to come up with a whole number for y. I keep coming up with fractions from $y = \frac{1}{x +2}$ I've tried numerous numbers, as in, $ 1, 2, 3 ,4 , 5, -1, -2, -3, -4, -5$. For example, $y = \...
1
vote
4answers
51 views

Solving the Inequality $\frac{14x}{x+1}<\frac{9x-30}{x-4}$

The question says to find all the integral values of x for which the inequality holds. the question is $$\frac{14x}{x+1}<\frac{9x-30}{x-4}$$ My Solution \begin{align} & \frac{14x}{x+1} < ...
0
votes
3answers
52 views

Maximize $k=x^2+y^2$ Subject to $x^2-4x+y^2+3=0$

Question Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$. Find the maximum and minimum values of $x^2+y^2$. My work Let $k=x^2+y^2$ Therefore, $x^2-4x+y^2+3=0$ ---> $k-...
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0answers
19 views

how do we get the given diophantine equation and it's value

\label{thm:3} The only squares of the form $${\overline{aa \ldots ab \ldots b}}_{(10)}$$ in decimal representation are the trivial infinite families $10^{2i},\; 4\cdot 10^{2i}$, $9\cdot 10^{2i}$ with $...
0
votes
0answers
39 views

theorem 2 of perfect powers with all equal digits but one

Can someone help me understand what happened to this equation from the paper entitled Perfect Powers With All Equal digits but one...I don't understand the part when it lets $a$ and $c$ not equal to ...
-6
votes
2answers
48 views

A and B are doing a job together. How long will take B alone to do the job? [closed]

A and B working together can do a job in 6 days.A become ill after 3 days of working with B, and B finished the job, continuing to work alone, in 9 more days. How long will take B alone to do the job?
2
votes
3answers
53 views

Showing a system of equations having two solutions in $\mathbb{R}^2$

Consider the system of equations in $\mathbb{R}^2$ \begin{align*} \xi^2 + \eta^2 &= 4\\ e^{\xi} + \eta &= 1 \end{align*} Show that the system has two solutions in $\mathbb{R}^2$ has two ...
1
vote
4answers
99 views

Why should $b$ groups of $a$ apples be the same as $a$ groups of $b$ apples?

Why should $b$ groups of $a$ apples be the same as $a$ groups of $b$ apples? We where taught this so it seems rather trivial but the more I think about it the more I feel that it is not. I'm trying ...
1
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4answers
78 views

Why don't parentheses matter in this case of multipication

Very basic question but can't seem to wrap my head around why this happens. Normally parentheses indicate that the operation inside must be carried out first. In this case: (a * a * a)*(a * a * a * ...
0
votes
1answer
54 views

Evaluate the limit of the form $\lim_{h\to 0} \frac{f(x_o+h)-f(x_0)}{h}$

Limits of the form $$\lim_{h\to 0} \frac{f(x_o+h)-f(x_0)}{h}$$ occur frequently in calculus. Evaluate this limit for the given $x_0$ and function $f$: $f(x)=3x-4, \ \ \ \ \ \ x_0=2$ Okay so I know ...
3
votes
4answers
59 views

The roots of $ax^2+bx+c$ are 6 and $P$. The roots of $cx^2+bx+a$ are $Q$ and $R$ what is the value of $P\times Q\times R$

Problem The roots of $ax^2+bx+c$ are 6 and $P$. The roots of $cx^2+bx+a$ are $Q$ and $R$ And we are asked to find $P\times Q\times R$ by using the identities: $P(x)=Q(x)\times D(x)+R(x)$ where $P(x)$ ...
1
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4answers
89 views

Find the limit $\lim_{t \to 9} \frac{3-\sqrt{t}}{9-t}$

Find the limit $$\lim_{t \to 9} \frac{3-\sqrt{t}}{9-t}$$ Here's what I have done: $$\lim_{t \to 9} \frac{3-\sqrt{t}}{9-t} \\ = \frac{3-\sqrt{t}}{9-t} \cdot \frac{3+\sqrt{t}}{3+\sqrt{t}} \\ =\frac {...
-2
votes
1answer
21 views

Inverse funtion's condition .

To have inverse function, there should be condition of $1-1$ and onto function. Such as $f(x)= x+5$. But what about $f(x)=\frac{3}{x-1}$ (Here $x$ isn't equal to $1$). How it can be a onto function. ...
2
votes
5answers
190 views

How to find this function, and what method to use? [closed]

The function is $f(x-\frac{1}{x})= x^3-\frac{1}{x^3}$ and they are asking us to find out what $f(-x)$ is?
1
vote
3answers
60 views

Can a quadratic have non-zero coefficients when the roots are $k$ and $-k$?

Can a polynomial like this: $$ax^2+bx+c$$ have two opposite roots, without either $b$ or $a$ being equal to zero?
9
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3answers
150 views

If $x^2+\frac{1}{2x}=\cos \theta$, evaluate $x^6+\frac{1}{2x^3}$.

If $x^2+\frac{1}{2x}=\cos \theta$, then find the value of $x^6+\frac{1}{2x^3}$. If we cube both sides, then we get $x^6+\frac{1}{8x^3}+\frac{3x}{2} \cdot \cos \theta=\cos ^3 \theta$ but how can we ...
0
votes
1answer
41 views

You own $19.75 in dimes and quarters - there are 100 coins in all - how many dimes?

I have been stuck on this problem for 30+ minutes and I can't seem to get the correct answer; there must be something that I am missing/doing wrong!! You own $19.75 in dimes and quarters There are ...
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votes
1answer
28 views

Solving equivalence relations

A relation R is defined on R such that aRb if and only if |a−b|< 1. Show that R is reflexive and symmetric but not transitive. Could you explain in detail? I tried reflexivity, but i don't know ...
2
votes
3answers
65 views

If $\sum_{n=0}^{\infty}|a_n|^p,\sum_{n=0}^{\infty}|b_n|^p $ converge then $\sum_{n=0}^{\infty}|a_n+b_n|^p$ converges

For a real $p\geq 1$, why if $\sum_{n=0}^{\infty}\left|a_n\right|^p,\sum_{n=0}^{\infty}\left|b_n\right|^p $ converge then $\sum_{n=0}^{\infty}\left|a_n+b_n\right|^p$ converges? I've tried using the ...
-1
votes
2answers
98 views

Trigonometry. If $A+B+C =180$ then find maximum value of $\sin^2(A) +\sin^2(B)+\sin^2(C)$ [closed]

If $A+B+C =180$ then find maximum value of $\sin^2(A) +\sin^2(B)+\sin^2(C)$
0
votes
4answers
49 views

calculus problem, arithmetic

Could someone help me with this please? I am solving a limit problem, the problem is not understanding what to do but only this small step on the way. I am missing how this can be, maybe I just have ...
3
votes
3answers
104 views

Solving the trig inequality $|\sin{x} + \cos{x}| > 1$

$|\sin{x} + \cos{x} |> 1$ How to solve this kind of question? Is there any websites to learn trigonometry inequalities? My teacher only taught us the simple question but not the complicated one. ...
1
vote
1answer
34 views

If A and B are positive real numbers and each of the equations: $x^2+ax+2b=0$ and $x^2+2bx+a=0$ has real roots, what is the smallest value of A+B

Problem In the equation:$x^2+ax+2b=0$ and $x^2+2bx+a=0$, we have to figure out the sum of a and b by using the following identity: $P(x)=Q(x)*D(x)+R(x)$ where P(x) is the equations and q(x) is the ...
0
votes
3answers
39 views

R and S are roots of $ax^2+bx+c=0$ so what is the value of $\frac{1}{r^2}+\frac{1}{s^2}$ in terms of a b and c

Problem The problems asks us in the equation $ax^2+bx+c=0$ and we need to compute $\frac{1}{r^2}+\frac{1}{s^2}$ in terms of a,b, and or c My steps In this problem we have to use the factor theorem ...
1
vote
1answer
56 views

Give an example of an inequality with exactly 3 solutions. [closed]

I am helping a friend try and do their pre-algebra and cannot for the life of me figure this out. I'm pretty sure it's an error in the phrasing, but I'm not too sure. The only inequalities I can come ...
2
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0answers
58 views

Discriminant of a polynomial definition

Why is the discriminant of a polynomial defined as the product of squared differences of roots? How do I intuitively understand it? Why was this definition chosen?
2
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1answer
47 views

Trying to prove that geometric and algebraic interpretations of the dot product are equal.

I am a Classics major trying to teach myself physics. I am on summer vacation at the moment, and I am going through a book called Classical Mechanics by J. Taylor. I am on the first chapter, and I ...
2
votes
1answer
33 views

Identify perfect square trinomial

I've read the definition perfect square for numbers, that is a number is a perfect square if it is the product of two equal integers. Now I'm studying perfect square trinomial so I'm confused about ...
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0answers
56 views

Which method do you use if you are faced with big equations?

I have a general question: Often, in applied mathematics, if a system is modeled by mathematical equation, these equations can become increase unwieldy. Here many mistakes are possible such as copy-...
9
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1answer
135 views

Is $n^7 - 77$ ever a Fibonacci number?

As the question title suggests, is $n^7 - 77$ ever a Fibonacci number, where $n$ is a integer?
2
votes
1answer
24 views

If $N = q^k n^2$ is an odd perfect number, is it possible to have $I(n^2) = I(q^k) + c$, for some constant $c > 0$?

The title says it all. If $N = q^k n^2$ is an odd perfect number, is it possible to have $I(n^2) = I(q^k) + c$, for some constant $c > 0$? Here $I(x)$ is defined to be the ratio $$I(x) = \...
0
votes
1answer
32 views

How to calculate the stake required to win $\$X$ at $Y$ odds

I want to cover a $\$100$ bet I have on the Hawthorn Hawks winning the Australian Football League grand final. If they lose the grand final, I lose my $\$100$. Another team, Geelong Cats, are ...
2
votes
2answers
60 views

How do you say: $\sqrt[z] x$ where $z > 3$?

My whole mathematics is in chaos right now.... I forgot how to say: $\sqrt[z] x$ and I don't know where else to ask - I know how to say ${d}\sqrt x$ - this is just: $d$ times the square root of $x$; ...
3
votes
2answers
56 views

Triangle with $3$ unknowns

I have a situation where I am trying to calculate a leading shot for a character in a 2D top down game. The enemy character moves with a certain speed $s$, which is applied to its normalized ...
3
votes
2answers
57 views

Find all $a, b, c, d$

Find all $a, b, c, d$ satisfying $$\frac{x^4+ax^3+bx^2-8x+4}{(x^2+cx+d)^2} = 1$$ My answers are: $$\begin{cases} d=2\\ d=-2 \end{cases} \quad \begin{cases} a=4\\ a=-4 \end{cases} \quad \begin{cases}...
0
votes
1answer
21 views

Creating a logarithmic function with known $x-y$ axis intersections

I know that to plot a straight line that intersects the axis in point $(x_1,y_1)$,(x_2,y_2)$ one can use this equation. $$(x_2-x_1)\cdot (y-y_1)=(y_2-y_1)·(x-x_1)$$ To be more specific if i want a ...
0
votes
1answer
31 views

Prove $(1+c/a+|b/a|)<0$ from $(a+b+c)(a-b+c)>0$

While doing a certain sum I got stuck at a step.I am getting $(a+b+c)(a-b+c)>0$.I need to prove $(1+c/a+|b/a|)<0$.Is it possible?How? a is not 0. The original question : If $ax^2+bx+c=0$ has ...
0
votes
0answers
15 views

identify notable (special) binomial products

I've been ask in homework to identify, among others, the following special binomial products: $(6x-2y)(2x-6y)$ $(x+3y)(7x^9+4)$ $(a^2b+c)(ac+b)$ I've been taught only 4 special binomial products $...
0
votes
0answers
17 views

Considerations for Gauss's lemma about polynomials

According to Gauss's lemma , a polynomial factorizable over $\Bbb{Q}$ is also factorizable over $\Bbb{Z}$.But how can we achieve such factorization?? For example how can we convert $p(x)=(x+\...
2
votes
2answers
31 views

A rectangle with perimeter of 100 has area at least of 500, within what bounds must the length of the rectangle lie?

Problem The problem states that there is a rectangle that has a perimeter of $100$ and an area of at least $500$ and it asks for the bounds of the length which can be given in interval notation or in ...
0
votes
1answer
26 views

Prove that $b^2/q^2=ac/pr$?

Let $\alpha_1,\alpha_2$ and $\beta_1,\beta_2$ be the roots of the equation $ax^2+bx+c=0$ and $px^2+qx+r=0$ respectively.If the system of equations $\alpha_1y+\alpha_2 z=0$ and $\beta_1 y+\beta_2 z=0$ ...
2
votes
2answers
67 views

Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$

As the title says I need to prove the following by induction: $$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$ When trying to prove that P(n+1) is true if P(n) is, then I ...