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
165 views

Solving System of Equations using transformation rotation

I've never had to post the same question twice, but my last post is getting filled out with work and I'm going about it a different way so I figured i'd try a whole different question So This is the ...
1
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1answer
40 views

Finding the alternate forms of ratios

This is a super basic algebra question but I can't figure out how you get the alternate form of: $$\frac ab = \frac cd$$ Which is: $$\frac {a+b}b = \frac{c+d}d$$ The process explaining how we ...
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2answers
53 views

Recursive Definition notation example

In my textbook, the author shows this example of recursion, and I can't make heads or tails of it. Can someone give a better explanation of this...
1
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2answers
66 views

Understanding $3^n < n!$

In my class, we are given the answer to this proof. I understand how the inequality was simplified, but don't understand why the bolded statement is true for $k+1,$ or more simply, how that proves by ...
1
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1answer
36 views

Prove that $\bar z_1 z_2+z_1 \bar z_2=2\Re(\bar z_1 z_2)$

Let us consider $z_1, z_2\in \mathbb C$; we have: $$\bar z_1 z_2+z_1 \bar z_2=2\Re(\bar z_1 z_2)$$ it is easy to prove if we put $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$. But suppose we do not want to use $...
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2answers
130 views

Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
1
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1answer
128 views

Looking for resources for understanding derivation of demand from utility

I am struggling with my homework and would very much appreciate a rundown of the math or pointers to where I can find help otherwise. Quoting from the assigment: There are $n$ sectors in the ...
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2answers
46 views

$2k^2+7k=2mk+3m+36$. Find all non-negative integer solutions.

I've tried this: $2k(k-m)+7k-3m-36=0$. And I'm stuck. How do I solve this one?
17
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2answers
444 views

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
3
votes
2answers
7k views

Exponential equations with variables on both sides

I have the following: $$8^{3x+4} = 5^{4x-2}$$ How would I solve this? I tried this: $$(3x+4)\log 8 = (4x-2)\log 5$$ but have no idea where to go from there. Thank you!
3
votes
2answers
154 views

Trigonometric equation $\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X)$

A trigonometric equation is to be solved, the solution ($X=10^\circ$) is very clear but I need a proper method $$\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X).$$
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1answer
61 views

Multiplying a factorial with non factorial

I'm trying to understand the following equation, do I multiply the 2 and the 1 to get (n+2)! ? $$(n+2)(n+1)! = (n+2)!$$
1
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2answers
72 views

$x^2-xy-2x+3y=11$. Find natural solutions.

I've got this factorizing: $(x-2)(x-y)=11-y$. And I'm stuck on it. How can I solve it?
0
votes
1answer
55 views

Adding and Subtracting numbers with exponents

Why is $2^{k+1} + 2^{k+1} = 2^{k+2}$ and not $4^{k+1}$
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1answer
25 views

$2x^2+2xy-x+y=112$. Find natural solutions.

I've got this and I'm stuck: $(x-2)(x-y)=11-y$. Is it possible to make something out of this? Oops, I've got that on the other problem.
1
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4answers
83 views

If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots $ n terms… [closed]

Question( from sequences) : If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots $ n terms = $(4000)(4000!)$ Then what is the value of n. How to proceed in this please suggest , will be of great help to ...
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2answers
44 views

What is the sum $\sum_{r=1}^\infty \frac{r}{4r^4+1}$ equal to?

Problem : If $$T_r =\frac{r}{4r^4+1}$$ then the value of $$\sum^{\infty}_{r=1} T_r$$ is ? How to start such problem I am not getting any clue on this please suggest thanks .
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2answers
59 views

The equation $5^x+2=17^y$ doesn't have solutions in $\mathbb{N}$

Problem: Prove that the equation $5^x+2=17^y$ doesn't have any solutions with $x,y$ in $\mathbb{N}$. I've been analyzing the remainder while dividing by $4$, but I'm getting nowhere.
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2answers
50 views

Evaluate this square root

$\sqrt{6 + 2\sqrt{5}} + \sqrt{6 - 2\sqrt{5}}$ I have no clue where to begin. I would appreciate a hint, the answer should be $2\sqrt{5}$ In general, how do you evaluate $\sqrt{a + b} + \sqrt{a - ...
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votes
6answers
121 views

Which of the following is equivalent to the expression? $i^{22}$

Which of the following is equivalent to the expression? $i^{22}$ A.) $-1$ B.) $i$ C.) $1$ D.) $-i$ What is $i$? How could it have a exponent if it's an imaginary number?
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6answers
51 views

Solve a linear function

How do I solve this homework assignment? For a linear function $y=f(x)$, $f(-3) = 25$ and $f(3) = 11$. Determine $f(-20)$. I know that with the values $f(-3) = 25$ and $f(3) = 11$ I am suppose to ...
2
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1answer
69 views

If $ab+bc+ca=1$, then $\frac{((a+b)^2+1)}{(c^2+2)}+\frac{((b+c)^2+1)}{(a^2+2)}+\frac{((c+a)^2+1)}{(b^2+2)} \geq 3$

Let $\displaystyle a, b, c> 0, ab+bc+ca=1$. Prove that the following inequality holds: $$\frac{((a+b)^2+1)}{(c^2+2)}+\frac{((b+c)^2+1)}{(a^2+2)}+\frac{((c+a)^2+1)}{(b^2+2)} \geq 3.$$ I tried ...
3
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2answers
119 views

Integral difficulties (attempt included)

I am having difficulties with the following integral. I began working on it and thought I had obtained the answer, but when I went to graph it I received an integral of 1. I obtained the same answer ...
1
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1answer
76 views

Functional equation- solving techniques

I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed ...
2
votes
2answers
109 views

Triangle inequality problem with equality

How does one prove that, for any reals $x,y$ , there holds the equality $$|x|+|y|+||x|-|y|| = |x-y|+|x+y|\quad?$$ I have tried this using both the reverse and triangle inequalities, but I cannot get ...
3
votes
1answer
84 views

How do I solve $x=\log^e{(x+1)}$ analytically?

How do I solve the following, analytically? $$x=\log^e{(x+1)}$$ It looks like it should be simple, but whether I take the $e$th root of each side or take the $\log$ of each side (ending up with a $\...
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votes
2answers
47 views

How would I be able to tell if some vector is in the span of a set of vectors?

Given the following, how would I be able to tell if b and c are in the span of the set of vectors S? Any help is appreciated. enter link description here
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1answer
37 views

A simple problem of the equation of a plane.

Two planes given $$x-y+z=5 , \hspace{0.5cm}x+y+z=3 $$ Their intersection is a line $l$.Find the equation of a plane such that the line $l$ is perpendicular to that required plane and this plane ...
0
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0answers
123 views

How to show that if average of squares equals square of average then all $X_i$ are equal?

Defining $A(X) = \sum_{i=1}^np_iX_i$, how would I show that given $A(X^2) = A(X)^2$, all $X_i$ must be equal? I tried by contrapositive - assume they are not equal and show that $A(X^2) \ne A(X)^2$. ...
4
votes
5answers
126 views

Show that $\gcd(a,b)>1$

Given are three natural numbers $a$, $b$ and $c$, for which $$\frac1a+\frac1b=\frac1c,$$ show that $\gcd(a,b)>1$. Could you someone provide a hint? I already tried algebraic manipulation, but ...
1
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2answers
58 views

integer $n$ for which $n^6+3n^5-5n^4-15n^3+4n^2+12n+3$ is a perfect Square

Prove that the no integer $n\;,$ for which $n^6+3n^5-5n^4-15n^3+4n^2+12n+3$ is a perfect Square. My Try:: We can write $(n^6+3n^5-5n^4-15n^3+4n^2+12n+3) = (n^3+an^2+bn+c)^2$ Now Here we have to ...
0
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1answer
20 views

How do I solve for x and y: x + 0.0467 + y = 1.000?

I am trying to find the isotopes for percent abundance question. I am looking over and answer and can't figure it out because of the math. Here it goes. 1) Set up a system of two equations in two ...
0
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1answer
54 views

find minimum and justify it

After having found the derivative of which if i am not mistaken is : I need to find the minimum of the function for which if I am not mistaken I equal to 0 the first derivative is this the right ...
2
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1answer
68 views

Partial fraction help

I need Help figuring out how to solve the indefinite integral of $$\int{ -5x^3-2x^2+32\over x^4-4x^3 } dx $$ using partial fractions. Please help. Thank you! I have already checked the online ...
0
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1answer
66 views

Value of $x$ where the graph lies below the graph of f(X)

From this question (How can I find the values of $x$ where a function lies below or above the axis?) I learn that "lies below" means $f(x)<0$ , now my question is , how can I check values that lie ...
0
votes
1answer
64 views

How can I find the values of $x$ where a function lies below or above the axis?

Let's imagine this problem: Find the values of $x$ where the graph of $$f(x)= \frac{3x^2}{x^2-1}$$ lies below the $x$-axis. I know how to find the intercept $(0,0)$, but I don't understand what ...
0
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3answers
39 views

no. of mapping from from $A\rightarrow B$ such that $f(i)<f(j)\;\forall \; i<j, $ is

If $A = \left\{1,2,3,4\right\}$ and $B = \left\{1,2,3,4,5\right\},$ Then $(a)\; :: $ Total no. of mapping from from $A\rightarrow B$ such that $f(i)<f(j)\;\forall \; i<j, $ is $(b)\;\;::$ ...
0
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1answer
71 views

Equation $ \sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}=\sqrt[n]{b_1}+\sqrt[n]{b_2}+\cdots+\sqrt[n]{b_l} $

Let $k,l$ be natural numbers and $\{ a_i, b_i \}$ be real positive numbers such that $a_1\leq a_2 \ldots \leq a_k$, $b_1\leq b_2 \ldots \leq b_l$ and $$ \sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{...
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4answers
61 views

Definite integrals involving $\ln x$

Alright, I have been working on this definite integral for the past couple days now and I can't for the life of me obtain the correct answer. I am not too sure where I am going wrong but I think the $\...
2
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2answers
68 views

Logarithmic function with strange bases

Given $\log_{4n} 40\sqrt{3} = \log_{3n} 45$, find $n$. I have rewritten $\log_{3n} 45$ as $\dfrac{\log_{4n}45}{\log_{4n}3n}$ and multiplied to get $\log_{4n} 40\sqrt{3}\cdot\log_{4n}3n = \log_{4n} ...
0
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1answer
41 views

Simplify complex fraction

I am working to simplify the equation 1+(1/x) divided by 1-(1/x), but i didnt get it. The solution given was to multiply by x/x and the answer is 1+(2/x-1) My solution was: (a) 1+(1/x) = (x+1)/x (b) ...
2
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1answer
35 views

Total Time Taken Question

Distance of chord = Time taken to "swim" to the desalination plant = I'm stuck here! The textbook working out is as such: I don't understand how they have the 'k' or 1/2 the runs river at 2 km ...
0
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1answer
29 views

Shortest Chord from origin to function

worked solution: Is this found using the distance of a line equation, where instead of co-ordinate points they use functions, so the two functions are g(x) and x (because the origin is on the line ...
1
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1answer
234 views

Arithmetic sequence of natural numbers

Consider an arithmetic progression of natural numbers with a non-zero common difference. For each of the members of the progression its square root is taken, and if the square root is not an integer, ...
0
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2answers
78 views

Finding all the roots (rational, irrational, and complex) of a polynomial

$$x^6-64$$ I have already tried using synthetic division but get stuck after the 3rd round of division. Then I tried looking at it as a difference of squares. That didn't clear anything up either. ...
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1answer
34 views

Solve for $x$ an inequality with logarithms [closed]

I am trying to solve this equation $\frac{(n(n-1))}{2} + X (2\log n +2) < nX $ I would like to solve it for X ? What should i do ? Thanks
4
votes
1answer
74 views

Algebra problem solve for a,b,c and d?

Can anyone find the values of these integers: a,b,c and d? $$1+\sqrt{2}+\sqrt{3}+\sqrt{6} = \sqrt{a+\sqrt{b+\sqrt{c+\sqrt{d}}}}$$ a+b+c+d = ? Thank you.
1
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2answers
65 views

Is my outcome wrong? (Evaluating a logarithm)

$$log\sqrt [ 4 ]{ x^2+y^2 } $$ $$log\sqrt { x+y } $$ $$logx^{ 1/2 }+log^{ 1/2 }$$ $$\frac { 1 }{ 2 }log (x+y)$$ The answer key saids: $$\frac { 1 }{ 4 } log(x^2+y^2)$$
1
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3answers
35 views

Not understanding what is going on in this problem (evaluating a logarithm)

$$\log({ \log }_{ 10 }10000)$$ Steps I took to solve this: ${ \log }_{ 10 }10000=4$ ${ \log }_{ 10 }4=y$ $10^{ y }=4$ ${ \log }10^{ y }=\log 4$ $y=\frac { \log 4 }{ \log 10 } $ doesn't seem to ...
0
votes
3answers
1k views

Write the equation of the tangent line of a circle

I'm totally lost with this question. I appreciate any kind of help. if the equation of a circle is $(x-3)^2+y^2=9$ Find : -Equation of the tangent line at $(2,2\sqrt2)$ -Equation of the tangent ...