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
4answers
2k views

Solve the equation $\sqrt{3x-2} +2-x=0$

Solve the equation: $$\sqrt{3x-2} +2-x=0$$ I squared both equations $$(\sqrt{3x-2})^2 (+2-x)^2= 0$$ I got $$3x-2 + 4 -4x + x^2$$ I then combined like terms $x^2 -1x +2$ However, that can not ...
2
votes
1answer
37 views

Linear independence of $(x\sin x)^{\frac{n-1}{2}}$ and $(x\sin x)^{\frac{n+1}{2}}$

Could you tell me why $(x\sin x)^{\frac{n-1}{2}}$ and $(x\sin x)^{\frac{n+1}{2}}$ are lineraly independent? I've tried $\alpha(x \sin x)^{\frac{n-1}{2}} + \beta (x\sin x)^{\frac{n+1}{2}} =0$ ...
1
vote
0answers
83 views

Matching numbers by $f(x)=\frac{1}{x}$

Let $0<x \leq 1$, We define a function such that $f(x)=y=\frac{1}{x}$ which results $y \geq 1$ . We have infinitely many numbers between $0$ and $1$, so we can match any $x$ to a number $y$ greater ...
3
votes
1answer
170 views

Question based on Triangle Inequality $\displaystyle |x+y|\leq |x|+|y|$

If $x,y,z\in \mathbb{R}-\left\{0\right\}$. Then prove that $\displaystyle 1\leq \frac{|x+y|}{|x|+|y|}+\frac{| y+z|}{| y |+| z |}+\frac{| z+x|}{| z |+| x |}\leq 3$ My Try:: Using Triangle Inequality ...
0
votes
0answers
34 views

Does $\sum_{k=0}^n x^k = \prod_{k=1}^n \left(x - \mathtt{i}^\frac{2 k}{n+1}\right)$?

This seems to be true: $$\sum_{k=0}^n x^k = \prod_{k=1}^n \left(x - e^\frac{2\pi i k}{n+1} \right)$$ but I don't know how to demonstrate it, and definitely not neatly. I'd like to see why it should ...
4
votes
4answers
15k views

How to factor a four term polynomial without grouping?

$$2x^3 + 9x^2 +7x -6$$ This equation doesn't factor by grouping, and other than that I have no idea how to solve this problem. Will someone please help?
4
votes
2answers
6k views

Trigonometry Airplane question. Finding bearing and distance.

A little background(if you don't care for my story, skip straight to the question): I've missed a few lectures from my teacher because I fell ill. Since I have no information to work with other than ...
1
vote
3answers
118 views

Solving the equation $\dfrac{(1+x)^{36} -1}{x} =20142.9/420$ for $x$.

How would one solve for x in the following equation: $\dfrac{(1+x)^{36} -1}{x} =20142.9/420$ I tried factorising the top but that didnt really help much. $((1+x)^{18} - 1)((1+x)^{18}+1)$ Any help ...
3
votes
2answers
77 views

How to define this pattern as $f(n)$

Given a binary table with n bits as follows: $$\begin{array}{cccc|l} 2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ ...
1
vote
1answer
67 views

The graph of $x^x$

I have a question about the graph of $f(x) = x^x$. How come the graph doesn't extend into the negative domain? Because, it is not as if the graph is undefined when $x=-5$. But according to the graph, ...
0
votes
1answer
484 views

Formula to increase/decrease a relative number based on a fixed number

Being a fairly good math student in high school, this is humbling. But my knowledge about graphs and formulas has greatly diminished since then. I'm trying to write a formula that calculates a ...
0
votes
4answers
331 views

rewriting equation in terms of $y$

From Stewart, Precalculus, 5th ed, P98, Q.45 $$x^2+xy+y^2=4$$ how can I re-write this equation in terms of $y$? I want to put this equation into graphing software but don't know to put $y$ on one ...
2
votes
4answers
265 views

Solve for $x$, $3\sqrt{x+13} = x+9$

Solve equation: $3\sqrt{x+13} = x+9$ I squared both sides and got $9 + x + 13 = x^2 + 18x + 81$ I then combined like terms $x^2 + 17x + 59 = 0$ I then used the quadratic equation $x= -\frac{17}2 ...
3
votes
4answers
128 views

Find $a,b,c \in \mathbb {Q}$

Find $a,b,c \in \mathbb {Q}$ such that: $\left\{\begin{array}{rl} x^3&\in \mathbb Q \\ x&\notin \mathbb{Q}\\ ax^2+bx+c &=0\end{array}\right.$ I tried Vieta's formulas, but seem like it ...
9
votes
2answers
565 views

Is it possible that a clock's three hands divide the clock face into 3 equal parts?

And is there a proof/disproof out of intuition? Here we ignore the structure of the clock and suppose the hands move with constant speed.
0
votes
2answers
993 views

Non-parallel vectors confusion

I've got a section in my textbook about non-parallel vectors, it says: For two non-parallel vectors a and b, if $\lambda a + \mu b = \alpha a + \beta b$ then $\lambda = \alpha $ and $\mu = \beta $ ...
4
votes
3answers
145 views

20 Gifts and four stacks, How many Gifts in each pile?

There are twenty gifts stacked up into $4$ piles. The first pile has $3$ less than the second pile. The second pile has $2$ more than the third pile. The fourth pile has twice as many as the second ...
1
vote
0answers
89 views

General solution for $M^{\circ -1 }(y)=x $ when $g(x)e^{f(x) }=y$

Reading this question $e^{C/x }-1=D/(x + a) $, i found my self completely unable to do anything. This is much more hard for me than my easy exercises about Lambert $W$-function. So I probably need ...
1
vote
3answers
296 views

Confusing math problem

How would I solve this question? I came across it and is really confused. The payment of Jon was bigger by $960$ than the payment of David. After the payment of David got increased by $10\%$, Jon and ...
0
votes
1answer
116 views

Easy way to check for a valid solution in this triple equality?

Let's say I have the following equalities $a_1x_1 + a_2x_2 + a_3x_3 + a_4x_4 = b_1x_1 + b_2x_2 + b_3x_3 + b_4x_4 = c_1x_1 + c_2x_2 + c_3x_3 + c_4x_4$ Where the $a$'s, $b$'s, and $c$'s are known, ...
1
vote
3answers
185 views

Getting rid of a floor function in the next expression:$\left\lfloor\frac{(x-2)^2}{4}\right\rfloor $, It is known x is odd.

I was wondering if there's a way in which you can get rid of a floor function in the next expression:$$\left\lfloor\frac{(x-2)^2}{4}\right\rfloor $$ It is known x is odd.
2
votes
2answers
130 views

Prove $\sin x +\frac{1}{(\sin x)^{\sin x}}<2, \quad x \in(0,\pi/2)$

No idea how to prove that. What should I try? $$\sin x +\frac{1}{(\sin x)^{\sin x}}<2, \quad x \in(0,\pi/2)$$
3
votes
1answer
55 views

How do I apply partial fraction expansion on $\dfrac{K}{(a+bz^{-1})(x+yz)}$?

I want to apply partial fraction expansion on $\dfrac{K}{(a+bz^{-1})(x+yz)}$. I'm not able to do it in the standard way, because one term has $z^{-1}$ term and the other has $z$. What is the approach ...
0
votes
4answers
254 views

How to graph the equation: $y=\frac {x-2}{x+1}$?

the title says it all. I'm pretty sure this is a hyperbola, but is there an alternative way of doing this besides a table of values? "Graph the equation $y=\frac {x-2}{x+1}$" I know that $x$ cannot ...
2
votes
2answers
221 views

Find the minimum values of $a,b,c,d,e,f$ that satisfy following equations

${ a }^{ 2 }+{ b }^{ 2 }={ c }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c \right) }^{ 2 }={ d }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c+d \right) }^{ 2 }={ e }^{ 2 }\\ { a }^{ 2 }+{ \left( b+c+d+e \right) }^{ 2 }={ ...
7
votes
2answers
221 views

If ${ x }^{ 4 }+{ y }^{ 2 }=1$ then $x$ and $y$ can be both rational numbers?

Can you give two numbers $(x,y)\in\mathbb{Q}$ such that ${ x }^{ 4 }+{ y }^{ 2 }=1$. I don't know if exists or not. I derive this equation questioning that if $\sin { \alpha } ={ x }^{ 2 }$ for ...
10
votes
8answers
550 views

How does one evaluate $\sqrt[3]{x + iy} + \sqrt[3]{x - iy}$?

The answer to a question I asked earlier today hinged on the fact that $\sqrt[3]{35 + 18i\sqrt{3}} + \sqrt[3]{35 - 18i\sqrt{3}} = 7$. How does one evaluate such expressions?
2
votes
1answer
56 views

For the following monic polynomial,$f$ of even degree how to prove that that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$

For any monic polynomial $f \in \mathbb {Q[x]}$ of even degree,how to prove, there exists polynomial $g \in \mathbb {Q[x]}$ such that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
2
votes
2answers
173 views

mixture problem

From Stewart, Precalculus, $5$th ed, p.$71$, q.$55$ The radiator in a car is filled with a solution of $60\%$ antifreeze and $40\%$ water. The manufacturer of the antifreeze suggests that, for summer ...
2
votes
8answers
230 views

What is $2 - 1 + 1$? [duplicate]

$2-1+1$; a fairly straightforward question, but I (well, not me, but Henry Reich) found something strange. Most people would evaluate it as $2+(-1)+1 = 2$; however, this goes against the famed, and ...
2
votes
2answers
289 views

Rational Expression Question. (Word problem)

Joe got a mark of $\dfrac{44}{50}$ on one test and $\dfrac{32}{x}$ on another test. If the average mark on the two tests was 80%, what value was the second test out of? My revised attempt: Still ...
0
votes
2answers
343 views

How to derive the sum of an arithmetic sequence?

I'm attempting to derive a formula for the sum of all elements of an arithmetic series, given the first term, the limiting term (the number that no number in the sequence is higher than), and the ...
5
votes
3answers
307 views

How many functions $f:\{1,2,3,4\}→\{1,2,3,4\}$ satisfy $f(1)=f(4)$?

I just need a hint or a way to think a about this problem: $f(1)$ can be $1, 2, 3, 4$ and $f(4)$ can be $1,2,3,4.$
3
votes
3answers
161 views

Function such that $f(x) = -1$ for $x < 0,$ and $f(x)=1$ for $x > 0$?

What is a function to returns $-1$ if number is negative, $1$ if positive, and zero if number is equal to 0? for example: $$ f(-8) = -1 $$ $$ f(8) = 1 $$ $$ f(0) = 0 $$ for $$x < 0$$ maybe? $$ ...
2
votes
3answers
86 views

How many functions $ f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ satisfy $f(1) + f(2) + \dots + f(10) = 2$?

How many functions $ f: \{1, 2, 3, \dots, 10\} \to \{0,1\}$ have this property: $$f(1) + f(2) + \dots + f(10) = 2.$$ I understand just $2$ functions can be $1$, the rest have to be $0$, in total ...
1
vote
2answers
303 views

How do I evaluate the following expression?

How to evaluate the following expression: $\displaystyle \frac{1}{\sqrt{2}+1}+ \frac{1}{\sqrt{3}+\sqrt{2}}+\frac{1}{\sqrt{4}+\sqrt{3}} +\cdots +\frac{1}{\sqrt{9}+\sqrt{8}}$
2
votes
0answers
66 views

Perhaps an easy algebra problem, but it still evades me

I need help spotting a corresponding transformation Let $x,y$ be some variables and $$z=z(x,y)$$. We have a transformation $X(\lambda):(x,y,z)\to (x',y',z')$, such that $$x'= x\exp(a\lambda)\\ ...
4
votes
1answer
114 views

Work-Time Problem

$10$ cats can eat $10$ mice in $20$ minutes. $2$ cats started eating $60$ mice in $3$ minutes, then another $6$ cats were added, how many more minutes will it take them to consume the remaining mice? ...
1
vote
1answer
418 views

Using Semi-circle find side of triangle

The figure below above shown a bicycle path. If semicircular portion $ABC$ is $100$ $\pi$ and $CD$ is $100$$ft$ then what is $AD$? I have tried to find the diamenter of the circle and the ...
4
votes
2answers
92 views

Simplifying this expression $(e^u-1)(e^u-e^l)$

Is it possible to write the following $$(e^u-1)(e^u-e^l)$$ as $$e^{f(u,l)}-1?$$
15
votes
4answers
759 views

Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$

Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid? I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
2
votes
1answer
72 views

Simplifying $\left|\left|\sqrt{-x^2}-1\right|-2\right|$

How do we simplify the expression $\left|\left|\sqrt{-x^2}-1\right|-2\right|$? This is very confusing. Do they cancel out and become just simply $\sqrt{-x^2}-1-2$?
0
votes
4answers
164 views

Completing the square with simple polys

I am suppose to rewrite $x^2 + x + 1$ by completing the square. I don't really know what that means but I know that if I add 3 at the end of this I get $$(x + 2) (x - 1) - 3$$ this is the same as ...
1
vote
4answers
92 views

Rationalizing quotients [duplicate]

I have $$\frac{\sqrt{10}}{\sqrt{5} - 2}$$ I have no idea what to do, I know that I can do some tricks with splitting square roots up but pulling out whole numbers like I know that $\sqrt{27}$ is ...
1
vote
3answers
104 views

Simplifying fractions with fractions

I am trying to simplify $$\frac{\frac{y}{x} - \frac{x}{y}}{\frac{1}{y} - \frac{1}{x}}$$ I make the top part into $$\frac{y^2 - x ^2}{xy}$$ I know the bottom can be rewritten to just be ...
1
vote
3answers
133 views

quadratic equation precalculus

from Stewart, Precalculus, 5th, p56, Q. 79 Find all real solutions of the equation $$\dfrac{x+5}{x-2}=\dfrac{5}{x+2}+\dfrac{28}{x^2-4}$$ my solution ...
1
vote
3answers
61 views

Solving Equation Using Algebraic Method

How to solve these equations using an algebraic method? I need to show my working, don't you do something in reverse, like 7 multiplies by something. I haven't done it in class. ...
0
votes
3answers
83 views

Which is true $A$ is subset of $B$ or $B$ is subset of $A$.

Consider the sets dened by the real solutions of the inequalities $$A=\{(x,y):x^2+y^4\le 1\}$$ and $$B=\{(x,y):x^4+y^6\le 1\}$$Then which is true $A$ is subset of $B$ or $B$ is subset of $A$. ...
1
vote
2answers
32 views

Finding $y$ value of canonical ellipse.

I have an ellipse: $$ \frac{x^2}{a^2}+\frac{y^2}{b^2}=1 $$ This may be a simple question, but my mind plays tricks on me at the moment; Which is the most efficient way if I have $x$, $a$ and $b$ ...
1
vote
1answer
258 views

no. of real roots of exponential equation in three questions

How Can i calculate no. of real roots of exponential equation in three questions (1) $2^x = 1+x^2$ (2) $2^x+3^x+4^x = x^2$ (3) $3^x+4^x+5^x = 1+x^2$ My Try:: (1) Let $f(x) = 1+x^2-2^x$ now ...