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
13 views

mathematical model

The question: $g$ varies directly with $f$ and inversely with $c$ and the square of $d$. So we have to setup the equation given that information. Looking at my notes a bit it seems like it might be ...
5
votes
0answers
159 views
+250

How can I construct a solution for this system of many inequalities?

Let there be types $\omega\in\{0,1\}^n$ drawn according to some probability distribution. Suppose that these types are relayed through some imperfect message service. Specifically, any type $\omega$'s ...
0
votes
2answers
72 views
+50

Find the locus of points M the difference of the squares of whose distances from two given points A and B is equal to a given value c.

Find the locus of points M the difference of the squares of whose distances from two given points A and B is equal to a given value c. For what values of c does the problem have a solution? I am ...
6
votes
2answers
62 views

Minimum of $ay+az+bz+bx+cx+cy$ with $ab+bc+ca=xy+yz+zx=1$

Let $a,b,c,x,y,z\in\mathbb{R}^+$, and $ab+bc+ca=xy+yz+zx=1$. What is the minimum value of $ay+az+bz+bx+cx+cy$? When $a=b=c=x=y=z=\dfrac{1}{\sqrt{3}}$, the desired value is $2$. When ...
5
votes
5answers
163 views

Show: $(x+y)^4 \leq 8(x^4 + y^4)$ Using Cauchy-Schwarz Inequality

I wish to show the following statement: $ \forall x,y \in \mathbb{R} $ $$ (x+y)^4 \leq 8(x^4 + y^4) $$ What is the scope for genralisaion? Edit: Apparently the above inequality can be shown ...
0
votes
1answer
20 views

Substituting Formulas

Substituting the formula for height of a tree in the formula for volume of a tree, the new formula for volume becomes ________________. A)V = (1/3)πr(kr2/3) B)V = (1/3)πr2 C)V = (1/3)πkr3 D)V = ...
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votes
2answers
20 views

System of Equations Given One Equation

7=3x+2y-z How many more equations would you need to solve x, y, and z? In which variables can the additional equations be? Give examples of equations that would help solve these variables. (Hint: ...
0
votes
1answer
10 views

Distance/measurement question! Help!

There is a question in a book that I am trying to solve. "A man usually rides his bike 1 kilometers per hour, yet the wind slows him to 6.76 kilometers for 26 minutes and 5.55 kilometers for 10. How ...
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votes
4answers
54 views

How can we find factorials in decimal form? [duplicate]

I've heard of factorials such as $5!$ and $3!$, which work like this: $5!=5\times4\times3\times2\times1=120$ and $3!=3\times2\times1=6$. At least this is what we get. Also, surprisingly, $0!=1$, but ...
-4
votes
2answers
31 views

Find its simplest form [closed]

Given $$ \psi(x)=x^2+5 $$ Simplify $$ \frac{\psi(x+h)-\psi(x)}{h}, h\not =0 $$
-1
votes
1answer
17 views

sketch the following functions stating the domain and range of each [closed]

sketch the following functions stating the domain and range of each:and sketch if possible a. $y = \sqrt{9-x^2}$ b. $y = \sqrt{x-1}$ c. $y = |2x|$ d. $y= 1/x-4$ e. $y= |2x|-1$ please help me ...
0
votes
1answer
34 views

Discriminant of Quadratic with circle

The circle $x^2 + (y - c)^2 = r^2$, where $c > 0$ and $r > 0$, lies inside the parabola $y = x^2$. The circle touches the parabola at exactly two points located symmetrically on opposite sides ...
-2
votes
3answers
35 views

Logarithmic equation with negative component [closed]

What would be the solution for this since it has negative RHS? $$7 \times 7^{8v + 4.3} - 4 = 9$$
0
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1answer
19 views

Solving an equation with variable in exponent

can anyone please assist in solving the following equation for $t$? I am having trouble once since I have some $t$ terms which are in the exponent and some which are not... ...
1
vote
1answer
37 views

Equation to zero confused

How do I equal to zero any equation ? Is there any guide for this? I'm so confused about this. Example $$ \cos(x-y)= xe^y$$ Or $$\ln\left( \sqrt{x^2+y^2} \right)= 4 - xy $$
0
votes
2answers
22 views

Write the expression in terms of $\log x$ and $\log y$.

$$\log\Big(\frac{x^3}{10y}\Big)$$ Write the above expression in terms of $\log x$ and $\log y$. To be honest, I'm really unsure as to how the final answer should look like. In other words, ...
-1
votes
2answers
38 views

Need help quick [closed]

A large novel is divided into four volume set. volume one starts of page-one from the pages being numbered consecutively from volume one through volume four. in other words, if the last page of volume ...
17
votes
4answers
2k views

Are the equations $2x - 2y = 11, x = y - 2$ unsolvable?

My 9th grade son had this math problem, which seemed unsolvable to me: $$2x - 2y = 11$$ $$x = y - 2$$ So we can use substitution to come up with: $$2(y - 2) - 2y = 11$$ Now distribute: $$2y - 4 ...
3
votes
2answers
48 views

How to solve the following equation? $\log_3\big(\log_x(\log_416)\big)=-1$.

$$\log_3\big(\log_x(\log_416)\big)=-1.$$ I am trying to solve this equation for $x$. This is what I have so far: $$\log_3(\log_x 2)= -1.$$ Okay, now I have this: log2 = (1/3)logx How do I isolate ...
-5
votes
2answers
54 views

Decompose into partial fractions [closed]

$$ \frac{10 x^3 - 15x^2 - 35 x}{x^2 - x - 6} $$
3
votes
1answer
52 views

Find the range of values of $d$ for which the cubic equation $x^3-8x^2+12x+d=0$ has exactly $3$ distinct real roots

$$x^3-8x^2+12x+d=0 $$ I have worked this using calculus by finding the stationary points. However this is part of a problem which was under number theory. So I am still trying for a solution using ...
6
votes
4answers
135 views

Solving $ (x+1)^{x^2-4x+3} = 1 $ for x [closed]

Consider $$ (x+1)^{x^2-4x+3} = 1 $$ I need some hints to aid me with to problem.
0
votes
0answers
12 views

Determine the tangents at the pole for $r = 1 - 2\cos(t)$

In a polar function, $r = 1 - 2\cos(t)$ what are the tangents at the pole, considering $t$ an angle? I am not sure what the pole is BUT! $x = \cos(t) - 2\cos^2(t)$ $y = \sin(t) - \sin(2t)$ ...
8
votes
1answer
195 views

Existence of a Polynomial

Does their exist a non-linear polynomial $P(x)$ such that for every rational number $y$ there exists a rational number $x$ such that $y=P(x)$?
0
votes
2answers
19 views

Pointwise Limits: Diagonal Sequence

Disclaimer This thread is meant to record. (For more details see: Answer own Question) For jeopardy it is written as question anyway. Have fun! :) Problem Consider the pointwise limit ...
1
vote
1answer
54 views

Stuck in finding an alternative to finding the hypotenuse of a right triangle (using Dickson's method)

I was trying an alternative to finding the hypotenuse of a right triangle using simple algebra and Dickson's model of a right triangle. I have a problem: Given $r + t$ and $r + s$, how do you find ...
1
vote
3answers
58 views

What is the solution set of $x-(1/x)\le 0$?

The above inequality is from my maths textbook which has + in middle. Can you also tell me a ebook which discusses equations. I am just familiar with normal inequality and a student of 9th standard. I ...
0
votes
1answer
20 views

simultaneous equations word problem

A stadium has two different types of seats, the lower section and the upper section. The lower section has 12 more rows than the upper section each row in the lower section has 40 seats each row in ...
0
votes
1answer
16 views

Calculating screen intersection points with lines

I am writing a game and for the life of me can't figure out the most efficient way to program this. I have a boss who comes into the middle of the screen. He then engages 4 lasers which extend well ...
5
votes
2answers
73 views

$a+b+c+d+e$ divides $a^5+b^5+c^5+d^5+e^5-5abcde$

Let $a,b,c,d,e$ be integers such that $a(b+c)+b(c+d)+c(d+e)+d(e+a)+e(a+b)=0$. Prove that $a+b+c+d+e$ divides $a^5+b^5+c^5+d^5+e^5-5abcde$. I'm reminded of the factorization ...
2
votes
1answer
35 views

Applying the law of sines

A pilot is flying over a straight highway. He determines the angles of depression to two mileposts, $5$ mi. apart, to be $32°$ and $48°$. (a) Find the distance of the plane from point $A$. (the point ...
2
votes
1answer
20 views

How to find the summation of a fraction?

$$\sum\limits_{i=400}^{2000} 2^{3 - 4k}/8^{2k + 3}$$ After trying to decompose it a little bit I ended up with 1/64 $\sum\limits_{i=400}^{2000} 1/8^{2k}2^{4k}$ But I can't really get past, this... ...
6
votes
4answers
149 views

If $(x+\sqrt{x^2 + 1})(y+\sqrt{y^2 + 1})=p$, find $x+y$

I was given this factorization problem and I tried many things, but couldn't solve it. Can someone, please, give me a hint? If $(x+\sqrt{x^2 + 1})(y+\sqrt{y^2 + 1})=p$, find $x+y$. Here $x, y$ ...
1
vote
1answer
13 views

GCD of polynomials by using Euclid's algorithm

Let $g = x^2 +6x -7$ and $f = x^4 - 1$. Find the GCD of $f$ and $g$. So I started by evaluating $f/g$ and the result is $q = x^2-6x+43, r = -300x+300$. I tried to follow the algorithm one step ...
0
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1answer
11 views

Shifting Sequences

Disclaimer This thread is meant to record. (For more details see: Answer own Question) For jeopardy it is written as question anyway. Have fun! :) Problem Mostly well-known it holds: ...
0
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0answers
21 views

Cyclic quadrilaterals

Given a cyclic quadrailateral $ABCD$ such that $AB=BC=7$ Let M be the point of intersection of the 2 diagonals. If $DM=5$ Find $BM$. For a quadrailateral to be cyclic that means the sum of opposite ...
0
votes
1answer
35 views

Very confusing solution

$4y^2+z^2-x+16y-4z-20=0$ So ive been solving it in this way. $4(y^2-4y)+(z^2-4z)-x-20=0$ Is it possible to delete the -20 with factoring these terms $4(y^2-4y)+(z^2-4z)$ in this way? ...
0
votes
1answer
18 views

If $C_0,C_1,C_2,\cdots C_n$ denotes the binomial coefficients in the expansion of $(1+x)^n$ then $\sum^n_{r=0}\sum^n_{s=0} (C_r+C_s)$ =?

Problem : If $C_0,C_1,C_2,\cdots C_n$ denotes the binomial coefficients in the expansion of $(1+x)^n$ then $\sum^n_{r=0}\sum^n_{s=0} (C_r+C_s)$ = ? Solution : We have : $\sum^n_{r=0}\sum^n_{s=0} ...
1
vote
1answer
151 views

How to solve: $x^4+x^2=1$

I solved $x^4+x^2+1=0$. But, the above one is hard. The equation is too hard for me to understand. Can anyone solve it? Please help.
0
votes
2answers
18 views

Least value of n that makes the nth derivative of the function non zero at $x=0$

The function f(x) is as follows. $f(x)=6\tan (x)(e^x-x-1)-3x^3-x^4-x^5$ If the $n^{th}$ derivative at $x=0$ is non zero then what should be the least value of n? My obvious approach was to ...
1
vote
3answers
43 views

Verify the identity $\cos^2x-\sin^2x = 2\cos^2x-1$

I am having problems understanding how to verify this identity. I am quite sure that it is to be solved using the Pythagorean identities but, alas, I'm not seeing what might otherwise be obvious. I ...
-3
votes
1answer
49 views

find the slope and graph of x= -4/9 [closed]

Find the slope and the graph of x=-4/9 ? what is the slope based on that equation? This the only problem in my assignment that i cant solve
4
votes
1answer
19 views

Application of the law of sines

The path of a satellite orbiting the earth causes it to pass directly over two tracking stations $A$ and $B$, which are 50 mi apart. When the satellite is on one side of the two stations, the angles ...
6
votes
4answers
347 views

Proof of Inequality using AM-GM

I just started doing AM-GM inequalities for the first time about two hours ago. In those two hours, I have completed exactly two problems. I am stuck on this third one! Here is the problem: If $a, b, ...
0
votes
1answer
27 views

In which correctly way should be the equation I come up with his in my mind.

So I have a equation $x^2 = 2y^2+3z^2$. And if I want to make it equal to 0 which one is the corretly way of doing it. $0 = -x^2+2y^2+3z^2$ or $x^2-2y^2-3z^2 = 0 $ I came up with this in my mind ...
3
votes
0answers
32 views

Calculation of $\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{m^2n}{3^n\left(m\cdot 3^n+n\cdot 3^m\right)}$ [duplicate]

Calculation of $\displaystyle \sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{m^2n}{3^n\left(m\cdot 3^n+n\cdot 3^m\right)}$. $\bf{My\; Try::}$ Let $\displaystyle S = ...
2
votes
1answer
28 views

$|xy-zw|\leq 1/4$ for $x+y+z+w=1$

Let $x+y+z+w=1$, $x,y,z,w\geq 0$. If $x=y=1/2,z=w=0$, we have $|xy-zw|=1/4$. Is it true that $|xy-zw|\leq 1/4$ always?
2
votes
1answer
43 views

How to algebraically solve for $e^s$?

$p(1-p-a)e^{s(1-p-a)} = (1-p)(p+a)e^{-s(p+a)}$ Solving for $e^s = \frac{(1-p)(p+a)}{p(1-p-a)}$ Assume $a,p$ are constants in Reals and $s$ is a variable. However I am not able to arrive to this ...
0
votes
2answers
40 views

Properties of logarithms $e^{\ln(x)}$ and corollaries

If $e^{\ln(x)} = x$, does $$e^{\ln(\tan(\pi x/2))} =\tan(\pi x/2)$$ If not, what does the function equal and why?
2
votes
1answer
42 views

Relationships between altitudes and medians of a triangle

Median $CM$ and altitude $CH$ of $\triangle ABC$ trisect $\angle ACB.$ Find $\angle ACB$ Ok so I drew the picture and I don't see how I can find the answer with no information given about the ...