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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1answer
20 views

Solving $[(1-λ)a^{-p} +λb^{-p}]^{\frac{-1}{p}}=C$ for the variable $b$

How can I approach solving $[(1-λ)a^{-p} +λb^{-p}]^{\frac{-1}{p}}=C$ for the variable $b$ When the given answer is $$b =λ^{\frac{1}{pc^{p(1-λ)a^{-p^{\frac{-1}{p}}}}}}$$ Could someone elaborate the ...
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2answers
55 views

Given $f(x)= 2x^2 -x+2$, find and simplify $\frac{f(x+h)-(x)}{h}$

My solution to this problem is $x^2+h-2/x(x=h)$ Am I right? Also, for a deeper understanding of this problem, one should always view the difference quotient as the input? Correct?
0
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2answers
64 views

Verification using Maclaurin series

The exercise asks to verify $$y(x) = x + \frac{2x^3}{3!} + \frac{24x^5}{5!} + ... = \frac{1}{2}\log\frac{1+x}{1-x}$$ by expanding $\displaystyle \frac{1}{2}\log\frac{1+x}{1-x}$ in Maclaurin's series ...
0
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1answer
158 views

Change base log formula?

Im trying to change the base log from ln to log with the following formula. y = a * ln⁡(x+c) + b The ln equation is: ...
1
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1answer
39 views

Solving $(1+ r/100)^t =2$ for the variable $r$

I have to solve the equation $(1+ r/100)^t =2$ for the variable $r$: I do have the end result $r=100(2^{1/2}-1)$, but the process it takes to reach this end result eludes me.
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2answers
68 views

Normal distribution result with weird result

I was given this question in class: $f(x) = -5x^2 + 30x -49$ It looked like a fairly normal quadratic equation at first glance until I put it into the quadratic formula and got the values of $x_1$ ...
2
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4answers
100 views

Contest problem involving primes and factorization

Prove that for any nonnegative integer $n$, the number $$5^{5^{n+1}}+ 5^{5^{n}}+1$$ is not prime. I want only some hints and the method to follow, but I don't need the full solution. Thanks.
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2answers
79 views

Is limits $\lim_{x\to\infty}{(x-2)^2\over2x+1}=\dfrac{1}{2}$?

$$\lim_{x\to\infty}{(x-2)^2\over2x+1}=\dfrac{1}{2}$$ I used an online calculator and it said it was actually $=\infty$ Here's how I calculate it: $$\lim_{x\to\infty}\dfrac{x^2+4-4x}{2x+1}=\lim_{x\to\...
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3answers
127 views

Solving for $y$ in terms of $x$

I have this equation $y^2 -2y = x^3 + 2x^2 + 2x + 4$ How to solve this for $y$ in terms of $x$, because we have also the term $-2y$ on the left side?
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2answers
88 views

Sin & Cos Equation/Relation

If sin(x) = 0.3, find cos(pi-x) how i would solve this: let x = sin-1(0.3) solve for cos(pi-[sin-1(0.3)]) Is there a way to solve this by hand? Is the above method wrong?
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1answer
28 views

Exponential and Logarithmic Functions $\log_3 (x+7) - \log_3 (x-7) = 3$

Having trouble solving this problem. Where have I gone wrong? $$\log_3 (x+7) - \log_3 (x-7) = 3$$ $$\log_3 \left(\frac{x+7}{x-7}\right) = 3$$ $$\frac{x+7}{x-7} = 27$$ $$x+7 = 27 (x-7)$$ $$x + 7 = ...
1
vote
2answers
36 views

Exponential and Logarithmic Functions - $\log_2(x^2-4x-28)= 2$

I got $x= \frac{45}{x-4} $ I know its wrong. Can someone show me how to solve. Thanks
1
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1answer
169 views

Help with this trigonometry problem?

Is there an easier way of doing this problem: A square tower stands upon a horizontal plane. From a point in this place from which three of its upper corners are visible their angular elevations ...
0
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1answer
204 views

Pre calc question about cabon dating

Carbon dating. the radioactive element carbon-14 has a half-life of 5750 years. The percentage of Carbon-14 present in the remains of organic matter can be used to date the age of the organic matter. ...
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2answers
1k views

Modeling with Exponential and Logarithmic Functions

Been at this problem for an hour and I can't seem to solve it. Any takers? A certain culture of the bacterium Streptococcus A initially has 12 bacteria and is observed to double every 1.5 hours. ...
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0answers
49 views

Descartes' Rule of Signs [duplicate]

I have read several places that Descartes' Rule of Signs was familiar to both Descartes and Newton, and that both considered it too "obvious" to merit a proof. I know how to prove it, but I would like ...
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3answers
59 views

$\log_3 100 - \log_3 18 - \log_3 50$ [closed]

$$ \log_3{100} - \log_3{18} - \log_3{50} $$ How do I solve?
0
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1answer
39 views

Figures tangent to other figures

Find all values of $d$ where the graph of the polynomial $x^2-2014x+d$ is tangent to the graph of $x^2+2014x+y^2+2014y=-4056196$. $4056196=2014^2=(2*1007)^2=4(1007)^2$ $(x+1007)^2 +(y+1007)^2=-4(...
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3answers
60 views

How do you solve a equation by converting to logarithm form for the problem$3e^{x-4} +2=83$?

$3e^{x-4} +2=83$ I understand converting logarithms but i dont understand how to convert e into log form and solve. help would be deeply appreciated.
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5answers
258 views

How do I find the y intercept for $P(x)= (x-1)^2(x-3)$?

I am sketching a graph of this polynomial $P(x)= (x-1)^2(x-3)$. I have the zeros, and I have made a sign diagram. Thus, I have all my values except for the $y$ intercept. How do I get it? If I ...
2
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2answers
80 views

Evaluating $\lim_{x\to1}\frac{x^2}{(x-1)^2}$

$\lim_{x\to1}\frac{x^2}{(x-1)^2}$ How do I calculate this limit? I can't use L'Hospital's rule here and dividing by $x^2$ doesn't work either. Thanks
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2answers
47 views

How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$

How can I prove that $2(\cos^6(x)-\sin^6(x))-3(\cos^4(x)+\sin^4(x))=-4\sin^6(x)-1$ I tried to factor and I got $2\cos^4(x)+(-2\sin^2(x)-3)(\cos^4(x)+\sin^4(x))$ but that doesn't lead me to my goal. ...
0
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1answer
102 views

Polynomial Functions and Their Graphs - Seeking Help for a Deeper Understanding

Match the the polynomial function with one of the graphs I - IV. Give reasons for your choice. 1) P(x) = x(x^2-4) I chose number III because it reminded me of a power function raised to the 3rd ...
2
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3answers
218 views

Reason what 2^k in a sigma will do

I am trying to solve the following calculation without a calculator: $$\sum_{k=0}^82^k{8\choose k}$$ The first part: $$\sum_{k=0}^8{8\choose k}$$ is equal to $2^8$. I already know that the answer ...
2
votes
1answer
392 views

Sketch the graph of the polynomial function: $S(x) = \frac{1}{2} (x-1)^3 +4.$

The question is: Sketch the graph of the the function by transforming the graph of an appropriate function of the form $y = x^n$. Indicate all x and y intercepts on the graph. I am really trying to ...
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3answers
48 views

Sketch the Graph of $-x^3+27$

Not sure if this is the right way to solve. The question ask to sketch the graph of each function by transforming the graph of an appropriate function of the form $y=x^n$. Indicate all $x$- and $y$- ...
1
vote
1answer
56 views

Equation involving cosine

I don't really know how to crack this one. Any help appreciated. $$\cos\left(d\sqrt{4-d^2}\right)=-\frac{d}2$$ Why is MSE forcing me to write more? Jeez.
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2answers
121 views

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$

If $\sqrt{x+y}+\sqrt{y+z}=\sqrt{x+z}$, then $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=?$ I really am stumped on this problem. I squared the first equation and found that $-y = \sqrt{(x+y)(y+z)}$. So ...
3
votes
5answers
85 views

How to find the coefficient of $x^8$ in $\prod\limits_{i=1}^{10}{\left(x-i\right)}$?

How to find the coefficient of $x^8$ in $(x-1) (x-2) . . .(x-10)$. Is there any general formula to solve this kind of problems?
0
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1answer
61 views

extrema of a polynomial

Say I'm given a general sextic equation $$ax^6+bx^5+cx^4+dx^3+ex^2+fx+g =0$$ I want to know how I can tell where the critical points (or least say how many there will be) will be since the Abel-...
0
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2answers
159 views

how do i write an equation in standard form by completing the square for $x^2 -9y^2-4x-18y=14$

I'm really having trouble with completeing squares i can solve for circles and ellipses but i can't seem to understand hyperbolas or parabolas, help would be deeply appreciated.
0
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3answers
276 views

Sketch the graph of the polynomial function $P(x)= x(x-3)(x+2)$

Sketch the graph of the polynomial function. Make sure your graph shows all the intercepts and exhibits the proper end behavior. $$P(x) = x(x-3)(x+2) $$ Should I ignore $x=0$ as a value?
0
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1answer
37 views

Using Zeros to Graph Polynomials - Loose Definition Needed for Comprehension

I am having a hard time understanding this definition. The definition states. "If P is a polynomial function, then "c" is called a zero of P if P(c) = 0. In other words, the zeros of P are the ...
3
votes
8answers
256 views

How to solve $x$ in $(x+1)^4+(x-1)^4=16$?

I'm trying my hand on these types of expressions. How to solve $x$ in $(x+1)^4+(x-1)^4=16$? Please write any idea you have, and try to keep it simple. Thanks.
1
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4answers
71 views

how to find the root in $x^3-x^2+6x+24$?

I'm trying my hand on these types of expressions. how to find the root in $x^3-x^2+6x+24$ ? please write any idea you have and easy ways please ! thanks.
0
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1answer
37 views

Reversing $\sum_0^kx\cdot2^i$

I'm trying to write an algorithm, and the easiest way I found for explaining the mathematical problem I'm facing is the following: Assume you have $x$ US dollars and you're gambling on a roulette. ...
0
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3answers
126 views

Working with percentages: if 30000 is left after donating 4%, 7%, 8% to various causes, what was the original wealth?

Recently in an aptitude test I came across a type of question as follows: A cat eats 75% rats in a room, it can go in all rooms, if 1 rat remains in the fourth room, how many rats where there ...
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3answers
95 views

Two Questions Regarding Polynomial Functions.

I am studying Polynomial functions and their Graphs. I am currently looking at the definition for a polynomial function and I am trying to arrive at a deeper understanding; thus, please excuse ...
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3answers
84 views

Quadratic Function is Given - x^2-4x+4. What are the x intercepts

I am having a problem finding the x intercepts. I have found the Standard form of the Quadratic (Vertex form) -(x+2)^2 + 8 [by the way, does is matter if the 8 is in front of the minus sign or in ...
13
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1answer
278 views

How to prove $\frac{1}{x}=\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+2\sqrt{\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}}$

Question: Let $a,b,c>0$ are give numbers and $x>0$, such that $$ \sqrt{\dfrac{a+b+c}{x}}=\sqrt{\dfrac{b+c+x}{a}}+\sqrt{\dfrac{c+a+x}{b}}+\sqrt{\dfrac{a+b+x}{c}} $$ show that $$ \...
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3answers
266 views

if $abc=1$, then $a^2+b^2+c^2\ge a+b+c$

This is supposed to be an application of AM-GM inequality. if $abc=1$, then the following holds true: $a^2+b^2+c^2\ge a+b+c$ First of all, $a^2+b^2+c^2\ge 3$ by a direct application of AM-GM....
2
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3answers
47 views

How do I find the $x$ intercepts for $-x^2-3x+3$

How do I find the $x$ intercepts? $-x^2-3x+3$? I converted the function into vertex form but I am stuck at $3/4= -(x^2+1.5)$. Can someone give me and idea what I can do?
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3answers
218 views

Work out linear factors

$x^3 + 2x^2- 9x -18 = (x^2 - a^2)(x+b)$ were $a$ and $b$ are integers. Work out the three linear factors of $x^3 + 2x^2 - 9x - 18$. What would I do to answer this? I don't have a clue where to start
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3answers
562 views

How to solve this equation? $y^4-2y^2-\sqrt{y+1}=0$

How to solve this equation? $y>0$ $$y^4-2y^2-\sqrt{y+1}=0$$ I found one solution that $y=\frac{1+ \sqrt5}{2} $ Can someone help me?
4
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1answer
42 views

Steel mixture homework problem

My problem is the following: In a particular steel mixture, $20 \%$ of it consists from chromium and $10\%$ from nickel. How much of this steel mixture can be produced, if we have $1200$kg of ...
0
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1answer
37 views

Find $t$ in $N = b \times g^t$.

The problem is the following: Find the value of $t$ in $N = b × g^t$. So for example "$512.000 = 2000 × 2^t$" I'm not really a mathematician so their may be a simple way or it could be hard.
0
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1answer
51 views

$f(x) = 2x^2 + 4x+3$, Express in Vertex Form: Give the Vertex and $y-$ and $x-$ intercepts

I can't find the $x$ intercepts! My vertex form is $2(x+1)^2 +1$ My $y-$intercept is $3$ and my vertex is $(-1,1)$ However, I am stuck trying to find the $x$ intercepts. I have $-\frac{1}{2} = (x+1)...
1
vote
2answers
124 views

Parabola problem involving depth.

The equation for the parabola, I found, is $$y^2=3.2x$$ What is the depth of the reflector? I'm getting $\dfrac {25}{3.2}$.
1
vote
1answer
60 views

I Completed the Square for x^2-6x, but I don't have the correct x intercepts. How do I find the roots?

x^2 -6x when converted into standard form is (x-3)^2 -9 It has a vertex of (3,-9) It has a y intercept of y = 0 However, I thought the x intercept would be +3. Yet, the book says the x intercepts ...
0
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3answers
45 views

Seeking a deeper Understanding of Inverse functions f(x) = 4 - x^2, x> or = 0

I am seeking a deeper understanding of inverse functions $$f(x) = 4 - x^2, x\geq0$$ How does the Domain affect how one evaluates the function?