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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2
votes
5answers
105 views

Solve the equation $x^{4}-2x^{3}+4x^{2}+6x-21=0$ [closed]

Solve the equation $$x^{4}-2x^{3}+4x^{2}+6x-21=0$$ given that two of its roots are equal in magnitude but opposite in sign. I don't know how to solve it. The roots are given as $\pm\sqrt{3},1\pm i\...
0
votes
5answers
71 views

Remainder Theorem - $(x+1)^{2015}$

This one just caught me without a clue. Find the remainder when $(x+1)^{2015}$ is divided by $x$. Assuming I don't use Binomial expansion. What are other alternatives?
0
votes
1answer
29 views

How do I determine the graph of functions involving radicals?

What is the explanation behind: the graph of $h(x)=\sqrt{4-x^2}$ is the upper half of the graph of $x^2+y^2=4$ the graph of $g(x)=-\sqrt{2-x}$ is the lower branch of the parabola $x=2-y^2$ I kind ...
1
vote
1answer
17 views

Transformation of Graph

Hello all, I tried to solve this transformation and my answer was $-(x+3)^3+2$ my reason for thinking: reflect cubic power, shift to the left $3$ units, move up $2$ units. $-(x+3)^3+2$ However, the ...
4
votes
2answers
75 views

Stuck proving that if $m$ and $n$ are perfect squares. Then $m+n+2\sqrt{mn}$ is also a perfect square.

I am relatively new to proofs and can't seem to figure out how to solve an exercise. I am trying to prove: Suppose that $m$ and $n$ are perfect squares. Then $m+n+2\sqrt{mn}$ is also a perfect ...
0
votes
1answer
29 views

Dividing Algebraic Expression

When the denominator and numerator are the same, why isn't this simplified as $1$? For example, suppose we have : $$\frac{4x+2y}{4x+2y},$$ why is this simplified as $4x+2y$? Surely any value you ...
0
votes
1answer
31 views

The statement “four times as many cars were bought as planes” as an equation

How do I translate the sentence Four times as many cars were bought as planes. into an equation? My teacher told me that this was $c = 4p$, $c$ representing cars and $p$ representing planes. I ...
2
votes
3answers
70 views

Finding roots of a Complex Polynomial

Question Given that x = 3 is a solution to $$ x^3 - (7+3i)x^2 + (16+15i)x - 6(2+3i) = 0 $$ find the other two solutions. What I have attempted; If $x = 3$ is a root then $(x-3)$ is a ...
10
votes
5answers
1k views

How to show this fraction is equal to 1/2?

I have the fraction: $$\frac{\left(2 \left(\frac {a}{\sqrt{2}}\right) + a \right) a} {2(1 + \sqrt{2})a^2}$$ Using Mathematica, I've found that this simplifies to $\frac{1}{2}$, but how did it achieve ...
1
vote
4answers
71 views

how to find $t$ from $2t^2-0.01t^4=100$?

how to find $t$, from $2t^2-0.01t^4=100$? I was guessing may be I can take $t^2$ common but if it is so so why cannot we take $t$ common in other cases? I mean, for example: $t^2+4t=-4$ why can we ...
2
votes
4answers
147 views

What is exactly the inverse of the function $f(x)=\frac{x}{1-x^2}$?

How do we find the inverse of the function $f:(-1,1)\to \Bbb R$ by $f(x)=\frac{x}{1-x^2}$? The problem has been posted here and the answer below says "To show that $f$ has a continuous inverse, you ...
2
votes
1answer
55 views

Complex numbers with equal modulus.

$a,b,c$ are complex numbers whose modulus are equal. If $a+bc,b+ca,c+ab$ are real then find the value of $abc$? We can assume three complex numbers and then make six equations but that seems a very ...
2
votes
1answer
17 views

Translating a worded question into a linear equation system. 2

A rower travels upstream at $6km/h$ and back to the starting place at $10km/h$. The total journey takes $48$ minutes. How far upstream did the rower go? Thank you in advance.
0
votes
1answer
42 views

Quintic Polynomial

I have a polynomial of degree five as ($x > 0$) $y = A x^5 + B x^4 + C x^3 + D x^2 + E x$, where $A$ is positive. I would like to find some sufficient conditions (inequalities) on coefficients ...
0
votes
2answers
26 views

Need to change variables in equations with cosh.

i have these five functions: $x=\tau \cosh(s)$ $q=\tau \sinh(s)$ $y= \sinh(s)$ $p= \cosh(s)$ $u= 1/2*\tau*\cosh(2s)+1/2*\tau$ I need to write $u$ in terms of $x$ and $y$ I know the answer is $u=...
-1
votes
2answers
105 views

Does $[0.9999…]=1$? [duplicate]

We all know that $0.99999...=1$ So does that imply $[0.99999...]=1?$ Or do we consider it as $0?$ My doubt is: any gif of the form $[0.xyz...]=0$. If $[0.99999...]=1$ won't that be contradicting? ...
2
votes
6answers
99 views

Solving the absolute value inequality $\big| \frac{x}{x + 4} \big| < 4$

I was given this question and asked to find $x$: $$\left| \frac{x}{x+4} \right|<4$$ I broke this into three pieces: $$ \left| \frac{x}{x+4} \right| = \left\{ \begin{array}{ll} \...
2
votes
1answer
67 views

Find the number of dissimilar terms in expansion of $\left(1+x\right)^{2012}+\left(1+x^2\right)^{2011}+\left(1+x^3\right)^{2010}$

Find the number of dissimilar terms in expansion of $\left(1+x\right)^{2012}+\left(1+x^2\right)^{2011}+\left(1+x^3\right)^{2010}$ My attempt: I tried using Principle of Inclusion and Exclusion: $n(A)...
0
votes
1answer
59 views

find the domain and range and sketch the graph of $g(x) = \sqrt{9 - x^2}$.

I've found the domain but how do we find the range and sketch graph. is there a way to find the range. Also the graph of square root is like $e^x$ but facing downwards towards $x$ axis while $x^2$ is ...
-1
votes
1answer
57 views

Compute average velocity of a particle over a given time interval [duplicate]

I am really stuck with my math problem, would appreciate the answer below. A particle moves on a line away from its initial position so that after $t$ hours it is $s = 4t^2 + t$ miles from its ...
-2
votes
2answers
47 views

Solving for a variable in an equality with combinations [closed]

The question is Solve for $n$ in the equation: $$\binom{n+2}{4}=6\binom{n}{2}$$
1
vote
1answer
20 views

Need help figuring out how to keep a ratio constant between two functions

I'm coding up some custom code for an RPG character sheet, and need some help finding a mathematical formula to use. Alright, so I have two sets of two variables. State 1: damageTaken1 and maxHP1 ...
3
votes
1answer
49 views

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$ As $r$ is the integer root of the equation $(x-a)(x-...
9
votes
9answers
1k views

Find the number of bicycles and tricycles [duplicate]

Help for my son. My math is a bit rusty and I'm trying to remember how to go about answering this question: "There are 3 times as many bicycles in the playground as there are tricycles. There is a ...
4
votes
0answers
71 views

If $f(n)= \binom{n}{0}a^{n-1}-..+(-1)^{n-1}\binom{n}{n-1}a^{0}$ ,Then $f(2007)+f(2008) $

If $\displaystyle a= \frac{1}{3^{223}}+1$ and $\displaystyle f(n)= \binom{n}{0}a^{n-1}-\binom{n}{1}a^{n-2}+...........+(-1)^{n-1}\binom{n}{n-1}a^{0}$ Then value of $f(2007)+f(2008) = $ $\bf{My\;...
3
votes
1answer
54 views

Factore $(x+1)(x+2)(x+3)(x+4)-35$

I need to factor $$(x+1)(x+2)(x+3)(x+4)-35$$ I know that the answer will be $$(x^2+5x+11)(x^2+5x-1)$$ I go out only $$(x^2+5x)(x^2+5x+10)-11$$ Help me.
1
vote
1answer
61 views

Solving logarithmic equations without calculator

Hi I am stuck on this question $$ \log_x 10= 5 (\log_{10} x) +4 $$ The answer key gives the solutions $x = 10^{1/5}$ and $x = 1/10$.
2
votes
1answer
51 views

$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6,\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}=8$,then find $\frac{a}{b}+\frac{c}{d}$

$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=6,\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}=8$,then find $\frac{a}{b}+\frac{c}{d}$ I have tried multiplying the two given equations $\frac{a}{...
1
vote
1answer
47 views

Factorials/Binomial Coefficients (Finding Integer Solutions)

Question There are many integer solutions to the equation $\begin{pmatrix}n\\r\\ \end{pmatrix} = \begin{pmatrix}n+1\\r-1\\ \end{pmatrix}$ including $n = r = 1$. Find an expression ...
2
votes
4answers
178 views

Second derivative of $x^3+y^3=1$ using implicit differentiation

I need to find the $D_x^2y$ of $x^3+y^3=1$ using implicit differentiation So, $$ x^3 + y^3 =1 \\ 3x^2+3y^2 \cdot D_xy = 0 \\ 3y^2 \cdot D_xy= -3x^2 \\ D_xy = - {x^2 \over y^2} $$ Now I need to ...
0
votes
0answers
34 views

Formula for Ratio and Scaling

I'm trying to make a formula to suit what I need but im having a hard time. This is the result of missed learning them back then when I was still a student. I need a formula that fits the situation. ...
-1
votes
1answer
35 views

How did they get $\frac{x+y}{x^2y^2} |x-y|$?

How did they get $\frac{x+y}{x^2y^2} |x-y|$? Shouldn't it be $\frac{x+y}{x^2y^2} |y-x|$?
0
votes
2answers
38 views

Sum of first $n$ positive integers to a positive power $p$

Consider the sum $$\sum_{i=1}^{n}i^{p}\text{ , }p \in \mathbb{Z}^+\text{.}$$ Using a method in Spivak's Calculus, it can be shown that $$(n+1)^{p+1}-1 = \sum_{k=0}^{p}\binom{p+1}{k}\left(1^{k}+2^{k}+\...
0
votes
3answers
64 views

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. [closed]

Find the value of $3^{\log_4(5)} - 5^{\log_4(3)}$. Is there any property that can help here?
-1
votes
3answers
46 views

$(x^2-5x+1)^2-(8x^2-40x+8)(y^2-3)(y^2+3)+16(y^8-18y^4+81) > 0$ for all values of the variables [closed]

Help me to prove $$(x^2-5x+1)^2-(8x^2-40x+8)(y^2-3)(y^2+3)+16(y^8-18y^4+81) > 0 \ \forall x, y$$
0
votes
4answers
150 views

What is the point of finding a limit? Does limit give us a real/exact value?

I wonder whether limit gives us an exact value. I mean look at this example: $$\lim_{x\rightarrow 2} \frac{x^2 - 4}{x - 2}$$ is 4 , right? But what if we do not use limit. Obviously at $x = 2$, it ...
0
votes
2answers
38 views

Are $x=-\frac{m}{n}$ and $-x=\frac{m}{n}$ the same?

I was wondering that is $x=-\frac{m}{n}$ same as $-x=\frac{m}{n}$ The question popped into my mind when had $x=-\frac{11}{14}$ or $-x=\frac{11}{14}$ as an anwser to one of my equations. Was the $...
9
votes
8answers
996 views

Examples of fallacies in arithmetic and/or algebra [closed]

I'm currently preparing for a talk to be delivered to a general audience, consisting primarily of undergraduate students from diverse majors. My proposed topic would be Examples of fallacies in ...
1
vote
3answers
58 views

Converting $(1+…+n)^2*(n+1)^3$ to $(2+…+2n)^2$

I'm currently going through Calculus by Spivak by myself, and came across a proof by induction requiring to prove $1^3+...+n^3 = (1+...+n)^2$ Naturally, to prove this, I need to somehow convert $(1+.....
0
votes
2answers
21 views

Angle of Revolution for a truck

I am doing a problem out of my textbook, which I don't understand. In my Alg2/Trig class, we are learning about angular speed, and linear speed in terms of angular speed. I can't figure out how to ...
1
vote
2answers
77 views

If $\frac{\sec^8 \theta}{a}+\frac{\tan^8 \theta}{b} = \frac{1}{a+b}\;,$ Then prove that $ab\leq 0$

If $\displaystyle \frac{\sec^8 \theta}{a}+\frac{\tan^8 \theta}{b} = \frac{1}{a+b}\;,$ Prove that $ab\leq 0$ $\bf{My\; Try::}$ I am Trying To solve it Using Inequality. Using $\bf{Cauchy\; Schwartz\...
0
votes
1answer
42 views

Dealing with division by zero in a particular limit?

$$\lim_{x\to 14} \left(\left(x^2-30\cdot x+225\right)^\frac{1}{x^2-29\cdot x +210}\right)$$ I've tried to simply this equation: $$\lim_{x\to 14} \left(\left(\left(x-15\right)^2\right)^\frac{1}{\left(...
1
vote
4answers
87 views

How does $(1+\sin x)\cos x =\cos x+\sin x \cos x$? I have a feeling i missed a basic fact

Okay, so i have problem: $\cos x+ \sin (2x) = 0$. When I searched for help on the internet, everyone seemed to use that $(1+\sin x)\cos x =\cos x+\sin x \cos x$. Is this a formula that I have missed ...
-1
votes
1answer
37 views

What are the vertical asymptotes for this function?

$F(x)=\frac{5x^2}{4x^2}+9 $ Okay so when i graphed this function there are no vertical asymptotes but why is that ? Becuase if u set the denominator equal to zero then u can solve and get the ...
-2
votes
2answers
27 views

Inequalities and modulus [closed]

If two numbers are less than a given number, how can we algebraically show that their difference is also less than the given number . Both numbers are greater than zero and in $\mathbb{Q}$.
3
votes
1answer
49 views

How do you find the value of $N$ given $P(N) = N+51$ and other information about the polynomial $P(x)$?

Problem: Let $P(x)$ be a polynomial with integer coefficients such that $P(21)=17$, $P(32)=-247$, $P(37)=33$. If $P(N) = N + 51$ for some positive integer $N$, then find $N$. I can't think of ...
-3
votes
1answer
46 views

What is inverse proportionality?

Translated from danish: This musical instrument consists of some pipes that have different lengths, and that each has its own tone. The frequency of the tone is inversely proportional to the pipes ...
2
votes
1answer
51 views

Solving this exponential

I'm trying to solve $$ e^\lambda (1-\lambda^2) = 1$$ I know it has a solution at $\lambda = 0$. How do I get the second solution? Wolfram Alpha gives something around 0.71, but doesn't show an ...
2
votes
1answer
214 views

Solving $\frac{x}{5}+\frac{x}{10}+\frac{x}{4}+75=x$

I had this equation in my mathbook (Pitkä SIGMA 1: Funktiot ja yhtälöt) $ \frac{x}{5}+\frac{x}{10}+\frac{x}{4}+75=x $ I first start with transforming 75 to a fraction $ \frac{x}{5}+\frac{x}{10}+\...
2
votes
1answer
48 views

If $f(3) = 5$, write an ordered pair that must be on the graph of $y=f(x+5)+1$ [closed]

Can anyone please help me to understand how to solve this problem? Thank you. If $f(3) = 5$, write an ordered pair that must be on the graph of $y=f(x+5)+1$.