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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2answers
44 views

Isolating $x$ in logarithmic functions

I'd just like to check that I'm using the right technique. I have the following function and I am trying to isolate $x$: $$ \ln(x) + \ln(x-1) = 1$$ I take $e^x$ of both sides: $$ x + x-1 = e$$ $$2x – ...
2
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1answer
64 views

Finding an inverse function

I'd like to make sure I'm doing things right, my answer looks a little funny. I have the following function: $$g(x) = 3 + x + e^x$$ I am trying to find $g^{-1}(x)$, so I replace $g(x)$ with $y$ and ...
0
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2answers
69 views

quotient of complex numbers?

so I was wondering if you have two different equations having denominators $2+i$ and $2-i$ respectively how came the denominator of the quotient in standard form is $5$ for both equations? I tought ...
1
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1answer
48 views

Converting a repeating base $7$ expression to a fraction

A question was given to me to convert a decimal in base seven to a fraction in base seven, where the base $7$ expression was $._7515151515\ldots$. I understand this would be $\frac 57 + \frac 1{49} + ...
1
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1answer
41 views

Finding the x-intercept of a straight line

My question today is whether or not the formula for the x-intersect has been discovered for any straight line on a graph. I have been working on this for a bit and I think I have discovered a formula ...
0
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3answers
45 views

What am I doing wrong in searching for the intersection point between 2 linear equations?

y=4x+5 y=3x-7 First I take equation 1 and set y to 0: 0 = 4x + 5 -5 = 4x -5/4 = x I get that the information above only ...
1
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1answer
34 views

Help with Evaluating a Logarithm

A precalculus text asks us to evaluate $\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{256}}{\sqrt[6]{32}}$ I do the following: $\log_{8}\dfrac{\sqrt{2}\cdot\sqrt[3]{(2^2)^3\cdot 2^2}}{\sqrt[6]{2^3\cdot 2^2}}$ ...
2
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2answers
60 views

Inequality involving multiple square roots

Wolfram alpha solves $\sqrt{x+1}\ge\sqrt{x+2}+\sqrt{x+3}$ for $x$, and answers $x=-2/3(3+\sqrt{3})$. How did it do it? Thanks!
0
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1answer
71 views

How to factor out a variable from an exponential?

I hope this question isn't completely newbish. Lets say you have the equation: $y\sin(x)+y+e^y=x$ $y$ can obviously be factored out as $y(\sin(x)+1)+e^y=x$ But can y also be factored out of the ...
0
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1answer
24 views

Ratios. In how much time could they complete the work? [duplicate]

If 20 students can clean 5 classrooms in 4 hours, so in how many hours can 40 students clean 40 classrooms?
3
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5answers
150 views

How to solve: $\frac{3·x-5}{8·x-2}<6$

I'm trying to solve $\frac{3x-5}{8x-2}<6$ ? I'm not sure which first step to take. I mean if I multiply both sides by $8x-2$ then I'm not sure if the sign would switch, as this could be positive ...
1
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3answers
78 views

Given a rational, how to find the integers whose quotient it is?

I haven't found an answer to this anywhere. Excluding brute force, given a rational $q$ in its decimal form ($1.47$, for example), is there a good algorithm to find integers $m$ and $n$ such that ...
2
votes
1answer
66 views

How to compute $\sum_{n=0}^\infty \frac{4^{n-1}}{(\pi -1)^{2n}}$?

I wrote as $\displaystyle\sum_{n=0}^\infty \frac{4^{n-1}}{(\pi ...
9
votes
2answers
357 views

If $f$ is a strictly increasing function with $f(f(x))=x^2+2$, then $f(3)=?$

Bdmo 2014 regionals(a tweaked version of question): If $f$ is a strictly increasing function over the reals with $f(f(x))=x^2+2$, then $f(3)=?$ Obviously,$f(3)=f(1)^2+2$ but I can't see where we ...
0
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1answer
29 views

Finding eigenvalues of a determinant

I have a question regarding finding the eigenvalues of this determinant. $$\small \left| \begin{matrix}\dfrac{3}{4} - \lambda & \dfrac{1}{3}\\ \dfrac{1}{4} & \dfrac{2}{3} - \lambda ...
1
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1answer
38 views

Prove that $f(A)\leq max(f(P),f(Q),f(R))$

Consider any $\bigtriangleup PQR$ in the $x-y$ plane. Let $f(x,y)=ax+by+c$ , where $a,b,c\in\mathbb{R}$. Let $A\in\mathbb{R^2}$ be any point in the interior or on the $\bigtriangleup PQR$. Prove that ...
2
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2answers
81 views

$f(x) = x^3 - x$ then $f(n)$ is multiple of 3

If $f(x) = x^3 - x$ then $f(n)$ is multiple of 3 for all integer $n$. First i tried $$f(n) = n^3-n=n(n+1)(n-1)\qquad\forall n\ .$$ When $x$ is an integer then at least one factor on the right is ...
9
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4answers
188 views

Number of digits of the number of digits of the number of digits of $2014^{2014}$

How would you solve that problem : What is the number of digits of the number of digits of the number of digits of $2014^{2014}$ ? (for instance the number of digits of $12345678901234567890$ is ...
2
votes
2answers
35 views

Systems of equations with three variables

Consider: $a + b = 5$ $2a + b + c = 4$ $a - b - c = 5$ I like to use substituiton for solving systems of equations, so I firstly look at equation 1 and solve for $a$ $a + b = 5$ $a = 5 - b$ I ...
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3answers
230 views

How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\ $?

How to prove $\displaystyle\sum_{n=0}^\infty \frac1{n!}=e\ $? I thought about it but I could not find a proof. Please give me some hints?
0
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2answers
39 views

Having trouble with fractions

If I want to solve the following equation for $a$: $$\frac{3a}{12} - \frac{4a + 8} { 12} = 9 $$ What should my first steps be? If I should cross multiply the two fractions on the LHS, but, if I do ...
4
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2answers
142 views

Evaluating $\sum_{n=1}^{99}\sin(n)$ [duplicate]

I'm looking for a trick, or a quick way to evaluate the sum $\displaystyle{\sum_{n=1}^{99}\sin(n)}$. I was thinking of applying a sum to product formula, but that doesn't seem to help the situation. ...
3
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1answer
818 views

Given an ellipse's center, focus and point, find its equation.

Given an ellipse's center is $(2,1)$, focus is (2,4) and point is (3,-3), we have Plug in center: $\frac{(x-2)^2}{a^2}+\frac{(y-1)^2}{b^2} = 1$ Use focus: $4^2=a^2-b^2$ $16=a^2-b^2$ Use point: ...
1
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0answers
61 views

Given equation of parabola, find vertex and directrix

Given that $x^2-bx+17-ay=0$ has vertex $(3,2)$, find the directrix and focus. My attempt is to make it into the form $(x-h)^2=4a(y-k)$ which has focus $(h,k+a)$ and directrix $y=k-a$. Is this right? ...
0
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1answer
93 views

Understanding 2012 AMC 12B #23

Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23$, $-21$, $-17$, and $-15$, and $Q(P(x))$ has zeros at $x=-59$,$-57$,$-51$ and $-49$. What is ...
-1
votes
1answer
455 views

The sum of three consecutive terms in a geometric progression, and of their squares

If the sum of the squares of three different terms in a geometric progression is $S^2$, and their sum is $AS$, then prove that square of $A$ must lie between $1/3$ and $3$, i.e., $1/3 < A^2 < ...
5
votes
1answer
196 views

Prove that $512^3 + 675^3 + 720^3$ is a composite number

We have to prove that the number $$N=512^3 + 675^3 + 720^3$$ is composite. I tried to use the identity $(a^3+b^3+c^3)=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)+3abc$ hoping to take out some common ...
0
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1answer
71 views

Faster method for partial fractions

Is there a way to apply the "cover-up" method when solving for fractions of the following type? $$\frac {2x}{(x+1)(x^2+1)^2}$$ The long way would be; $$\frac {2x}{(x+1)(x^2+1)^2} = \frac {A}{x+1} + ...
3
votes
2answers
203 views

Calculate the value of $\sum\limits _{n=1}^{\infty }\:\dfrac{n}{2^n}$ [closed]

In a previous question it is asked to represent $f(x)=\dfrac{x}{1-x^2}$ as a power series. It gave me $\displaystyle\sum _{n=1}^{\infty \:}x\left(2x^2-x^4\right)^{n-1}$. Then they ask to use the last ...
1
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0answers
39 views

How to get the solution of sum of exponential?

I have an equation of the form: for $i\neq j$, $b_i\neq b_j$, $$ \sum_{k=1}^{n}a_ke^{-xb_k}=0. $$ I would like to get $x$. How to proceed? For $n=2$, the problem is fairly easy. I am interesting in ...
0
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3answers
42 views

How do I deal with algebraic exponents?

Can someone introduce me to algebraic exponents, to the likes of $8^{x+1} = 4^x$ or $3^{2x-1} = 27$? Google examples were a bit complicated for me, I'm hoping for some intuition here.
1
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2answers
43 views

implicit differentiation-trouble with algebra

I'm having trouble figuring out where to go with this implicit differentiation problem. Problem: Find $\frac{dy}{dx}$ given that $\sin(x)=e^{-y\cos(x)}$ Here is how I start: ...
0
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2answers
77 views

How do I determine if these functions are odd or even, or neither?

$$y=\frac{x^3}{x^4-x^2}$$ $$y=\frac{1}{x^3-1}$$ $$f(x)=\frac{3}{x^2-4}$$ $$y=\frac{x-3}{x+3}$$ $$f(x)=\frac{x^3}{x^5-x^2}$$ I substituted in $-x$ for each $x$, and got my result - however, it is ...
2
votes
3answers
266 views

Having trouble graphing circles and semi circles

I was asked to graph $y = \sqrt{1-x^2}$. How do I do that? I tried to do a table of values but that did not turn out correct Is their a trick to graphing circles, semi circles, hyperbolas etc
2
votes
3answers
58 views

Show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f$ is discontinuous at $c$

How to show $\lim \limits_{x \rightarrow c_{-}} f(x) \neq \lim \limits_{x \rightarrow c_{+}} f(x)$ imply $f: \mathbb R \rightarrow \mathbb R$ is discontinuous at $c$ ? I know that $f$ cannot have ...
1
vote
1answer
58 views

Solving $F(x) = 0.3$ where $F(x) = 1-\frac{200^{ 2.5 }}{ x^{2.5}}$

Consider the function $$F(x) = 1-\frac{200^{ 2.5 }}{ x^{2.5}}.$$ We want to solve $F(x) = 0.3$. Since $ F(x) = 0.3$ then we can say, $2.5 \ln (\frac{200}{x}) = 0.7$. $\ln(\frac{200}{x}) = 0 .28$. ...
0
votes
2answers
22 views

What is wrong with this technique for proving skew lines?

We have the following set of lines: $$L_1: \frac{x-2}{1}=\frac{y-3}{-2}=\frac{z-1}{-3}$$ $$L_2:\frac{x-3}{1}=\frac{y+4}{3}=\frac{z-2}{-7}$$ This leads to the following parametric equations: ...
0
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4answers
49 views

Integral values of $x$ for which the expression $x^2+19x+92$ is a perfect square

Calculation of Integral values of $x$ for which the expression $x^2+19x+92$ is a perfect square. $\bf{My\; Solution::}$ Let $x^2+19x+92 = k^2\;,$ where $x,k\in \mathbb{Z}$ $$x^2+19x+(92-k^2)=0$$ ...
0
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2answers
89 views

Why is $(1 - \frac{1}{p})^n$ close to $e^{-\frac{n}{p}}$ when $n$ and $p$ are large?

Looking at this answer by Henry birthday problem - expected number of collisions and struggling to figure out why it matches this other formula provided to me on a programming related question. ...
0
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3answers
125 views

what does slope mean in mathematics ?

I need your help. My question is: what does "slope" mean in mathematics? A clear example would be appreciated. Thank you.
4
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1answer
86 views

Showing that $|x-y| \leq |x| +|y|$ for $x.y \in \mathbb{R}$.

I know from intuition that $|x-y| \leq |x| +|y|$ for $x.y \in \mathbb{R}$. The way I would prove it is to use the triangle inequality: $|x-y| = |x+(-y)| \leq |x| +|-y| = |x|+|y|$ for $x.y \in ...
0
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5answers
99 views

Use induction to show that $3^n >n^3$ for $n≥4$

Use induction to show that $3^n >n^3$ for $n≥4$. (Note that you have to start at $n=4$ as the result isn't true for $n=3$ !) I am very new to using induction, but as I understand it I have ...
0
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2answers
430 views

Why do trigonometric equations have infinitely many solutions?

If i'm asked to solve the $\cos (\theta) =\frac{1}{2}$ why is the answer usually given as a general formula; in this case by: $\theta = \frac{5\pi}{3} + 2\pi k$ and $\theta = \frac{\pi}{3} + 2\pi ...
0
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1answer
25 views

Isomorphy of the integer and the polynomials

Find an Isomorphism from $$\Bbb{Z}[x]$$ to the ring of the Integer. Any help would be much appreciated!
6
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2answers
296 views

Find the maximum possible value.

For all ordered triples $(p,q,r)$ define the polynomial $$f_{p,q,r}(x)=x^3-px^2+qx-r$$ Let $a_{1},a_{2},a_{3},b_{1},b_{2},b_{3},c_{1},c_{2},c_{3}$ be (not necessarily distinct) positive reals such ...
1
vote
4answers
89 views

How is $2\sum_{n=2}^{\infty}\frac{1}{(n-1)(n+1)}=\frac{6}{4}$ calculated?

$$2\sum_{n=2}^{\infty}\frac{1}{(n-1)(n+1)}=\frac{6}{4}$$ I cant figure out why this is $\frac64$. I try to use telescopic series without success.
4
votes
5answers
498 views

Prove infinite series

$$ \frac{1}{x}+\frac{2}{x^2} + \frac{3}{x^3} + \frac{4}{x^4} + \cdots =\frac{x}{(x-1)^2} $$ I can feel it. I can't prove it. I have tested it, and it seems to work. Domain-wise, I think it might be ...
2
votes
3answers
73 views

How to prove that $\frac{a^n}{a^m}$ is equal to $a^{n-m}$? [closed]

How to prove that $\dfrac{a^n}{a^m}$ is equal to $a^{n-m}$? Thank you in advance.
1
vote
1answer
39 views

How many equations of this inequality?

What is the other equation for this inequality? $|2x - 7| \ge 1$ Is it $|2x - 7| \ge -1$ or $|2x - 7| \le -1$
1
vote
3answers
52 views

Simultaneous equations

I keep getting the following equation wrong: Firstly, I solve for y = x - 4, and substitute it in the second equation. Then once I get x from the second equation, I substite it back into the first ...