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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5
votes
2answers
161 views

How many times can a $4^{th}$ degree polynomial be equal to a prime number?

If $f(x)$ is a $4^{th}$ degree polynomial with integer coefficients, what is the largest set ${x_1, x_2, x_3, ...x_n}$ (where $x_i$ are integers) for which $|f(x_i)|$ is a prime number? Things I ...
2
votes
1answer
37 views

Stuck on rearranging of this equation

I need to get from $[(1-p)f+p(1-f)](1+v)-[(1-p)(1-f)+pf] = x$ to $(2+v)(f+p-2pf)-1 = x$ but I'm stuck. I'd appreciate any tips on what I should I do after the following. $(f+p-2pf)(1+v) + (f + p ...
-2
votes
1answer
89 views

$\cos x$, $\cos \pi x$, $|\cos \pi x|$ [closed]

Why is the graph of $\;\cos\pi x\;$ like a linear function? And $\left|\;\cos\pi x\;\right|$ like an absolute value function? I know about the absolute function, but isn't it the graph of $\cos x$ ...
1
vote
1answer
45 views

Smallest value of n for two algoritms with a certain running time

If one algorithm has a running time of $100n^2$ and another of $2^n$; how can I find the smallest value of $n$ such that the former is faster than the latter? I could do: $100n^2 < 2^n$ then ...
0
votes
1answer
58 views

Extract time frames from days

I am a computer programmer, and I like to performe some maths and I am not sure for the correct method to use. More specific, I am creating an application that charge a client based on time usage of ...
1
vote
3answers
600 views

Solve the following equation: $x^4- 2x^2 +8x-3=0$

Solve the following equation: $$x^4- 2x^2 +8x-3=0$$ We get 4 equations with 4 variables. But that is too difficult to solve. My try: Let $a,b,c,d$ be the roots of the equation. $$a+b+c+d=0$$ ...
1
vote
2answers
42 views

How to find the maxima?

This is a simple question : Find the maximum value of $\frac { 1 }{ { x }^{ 2 }-6x+2 }$ I rewrote ${ x }^{ 2 }-6x+2$ as $(x-3)^{2} - 7$, now when this is min, the original function is max, thus the ...
6
votes
4answers
731 views

If $a+\sqrt{b}=c+\sqrt{d}$ does $a=c$ and $b=d$?

If $a+\sqrt{b}=c+\sqrt{d}$ does $a=c$ and $b=d$? I am grading some problems and I don't think this true, but all of a sudden I am doubting myself...
0
votes
2answers
41 views

Simple algebra loss calculation

Kylie bought an item for $x$ and sold it for \$10.56. If Kylie incurred a loss of $x$ percent, find $x$. The answer is apparently "12 or 88" but I cannot see how they got there. I have tried ...
2
votes
2answers
136 views

Prove an inequality concerning $\sqrt[3]{4a^3+4b^3}+\sqrt[3]{4b^3+4c^3}+\sqrt[3]{4c^3+4a^3}$

Let $a,b,c$ be positive. I need to prove $\sqrt[3]{4a^3+4b^3}+\sqrt[3]{4b^3+4c^3}+\sqrt[3]{4c^3+4a^3}\leq \dfrac{4a^2}{a+b}+\dfrac{4b^2}{b+c}+\dfrac{4c^2}{c+a}$ Thanks!
0
votes
1answer
49 views

Searching for the ratio in alloys

Two alloys A and B are composed of two basic elements. The ratios of two compositions of two basic elements in the two alloys are 4:3, 5:4 respectively. A new alloy X is formed by mixing the two ...
0
votes
2answers
55 views

How to determine solutions: $\;2^x=3^y=36^{-z}\; \implies \frac1x +\frac 1y +\frac 1{2z} = \quad ?$

If $$2^x=3^y=36^{-z}\;$$ then $$\frac1x +\frac 1y +\frac 1{2z}$$ is equal to a) $\;0$ b) $\;1$ c) $\;-1$ d) none of theses How to solve this problemplease explain it
0
votes
1answer
70 views

Pythagorean motion

At the same moment two particles start respectively from vertices $B$ and $C$ of triangle $ABC$ which has a right angle at $C$. The particles move at constant speeds and arrive at vertex $A$ at the ...
1
vote
1answer
112 views

Re-arranging the equation $L=\sqrt{a^2\sin^2t+b^2\cos^2t\,}$ to find $\left(t\right)$?

How can I re-array the equation $L=\sqrt{a^2\sin^2t+b^2\cos^2t\,}$ to find the equation of $\left(t\right)$ ? $t=\,?$ I tried to solve it but I'm stuck at: $L^2=a^2\sin^2t+b^2\cos^2t$ ...
0
votes
1answer
99 views

Re-arranging the equation $t=\arctan\left(\frac{a}{b}\tan\theta\right)$ to find $\theta$

How can I re-array the equation $t=\arctan\left(\frac{a}{b}\tan \theta\right)$ to find the equation of $\theta$. $\theta=\,?$ Actually I tried this equation: ...
0
votes
1answer
57 views

Is it possible to find the term (variable) $c$ from the equation $a^2=\sqrt{b^2+c^2}$

If I have the equation $a^2=\sqrt{b^2+c^2}$. Is it possible to me to find the term (variable) $c$ from it ? $c=\,?$
0
votes
1answer
56 views

Any way to tell when an algebraic expression takes on values that are a square?

Say I have the expression $256x^2 -480x$. As a polynomial this isn't a perfect square. However that doesn't stop it from taking real values that are a perfect square for given x, such as x = 8. Is ...
1
vote
2answers
77 views

If $A=(-4,0)$ and $B=(4,0)$, what is the locus of points $P$ such that $|AP-BP|=16$? Does it even exist?

I am stuck in this question for about a week: If there are points $A$ and $B$ such that $A(-4,0)$ and $B(4,0)$ then what is the locus of points $P$ such that $|AP-BP|=16$? I think this is a ...
1
vote
1answer
70 views

Find the maximum of $xy(72-3x-4y)$?

$x$ and $y$ are positive. I have been stuck on this problem for a while now, any hints please?
0
votes
0answers
29 views

how to find the composite of a convex function with a linear map?

Let $g:F\to R$ be a convex function and $A:E\to F$ be a linear map, where E and F are Euclidean spaces. My question is in two parts: 1- How can we find the composite of the convex function g with ...
1
vote
1answer
77 views

Showing $\max\limits_{|z|=r}|p(z)| \ge |a_n|r^n$, without Cauchy integral formula.

Let $p(z) = a_n z^n + a_{n-1}z^{n-1} + \cdots + a_0$. My question is: Is there an elementary way to show that for all $r > 0$ $$ \max \limits _{|z| = r} |p(z)| \ge |a_n|r^n$$ without using ...
4
votes
3answers
133 views

$f(x)=x^{x}$ what happens when $x$ is a negative irrational number?

Just looking at negative numbers, $x^{x}$ is defined for all rational numbers (on the real plane) in all instances except whenever $x=\large \frac {2a+1}{2b}$ where $(a, b)$ are integers . However, ...
3
votes
1answer
145 views

Prove that $\frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2}\geq 1$

Let $abcd=1$ and $a,b,c$ and $d$ are all positive. Prove that $\dfrac{1}{(1+a)^2}+\dfrac{1}{(1+b)^2}+\dfrac{1}{(1+c)^2}+\dfrac{1}{(1+d)^2}\geq 1$ I am probably able to do this by assuming $a\geq ...
3
votes
1answer
47 views

How can I transform this equation in a conical?

In this equation $$2x²+y²-4x-6y+11=0$$ I got the result $(1,3)$ completing squares $2(x - 1)² + (y - 3)² = 0$   But on my list exercises, demanded that determine the foci, straight guideline ...
5
votes
1answer
85 views

Proving that an algebraic expression cannot be a square

Say that I have an expression in several variables, like $zxy+z^5x^2y^2 + xy + 24z$. To prove that it's not a perfect square, I write it in terms of one of the variables, say $x$. This makes it a ...
1
vote
1answer
54 views

Simplify Mathematical Expression

Can someone help me to simplify the following expression? I can assume b is small and $0<b<1$. $(C^{N}_{i})$ is the binomial coefficient. $$A=[(1-b)^{N} + \sum^{N}_{i=1} (C^{N}_{i} - 2 ...
2
votes
1answer
95 views

Find the value of $x+y$ if $x^2+y^2+10 = 2\sqrt2x+4\sqrt2y$

If $$x^2+y^2+10 = 2\sqrt{2}x+4\sqrt{2}y$$ then the value of $(x+y)$ is: a) $4\sqrt{2}$ b) $3\sqrt{2}$ c) $6\sqrt{2}$ d) $9\sqrt{2}$ Please teach me its basics and how to solve it?
1
vote
1answer
84 views

Is there an intuitive explanation for the formula for the number of observations in an average given two averages and a marginal observation?

First, apologies for the long-winded title! I'm helping my 10 year old son with math, and we had a set of problems based on the following scenario: Given an average of a set, a single marginal value ...
0
votes
1answer
1k views

Problem to find the intersection of a exponential and linear function

I have the problem to find the intersection of a exponential and linear function. My math teacher can't help me, but I'm interested how I can solve this. I tried to use the equating method, but it ...
1
vote
2answers
205 views

Show that if the roots of the equation $(a^2+b^2)x^2 + 2x(ac+bd) +c^2+d^2$ are real, they are equal

Please help. I am approaching it through the discriminant way, but I am struck. I TRIED IN THE FOLLOWING WAY: $$D\geq 0 $$
0
votes
2answers
98 views

Rearranging an inequality

We are given that $x > y \geq 0$ and we know $y > x/2$. I'd like to show that $x -y < x/2$? Here's where I've attempted so far and I am getting stuck. $x > y > x/2$ $x - y > 0 ...
0
votes
3answers
116 views

Determine the lengths of the sides of a right triangle

The positive real numbers $a,b,c$ are such that $a^2+b^2=c^2$, $c=b^2/a$ and $b-a=1$. Determine $a,b,c.$
2
votes
2answers
333 views

The golden ratio and a right triangle

Assume the square of the hypotenuse of a right triangle is equal to its perimeter and one of its legs is $1$ plus its inradius(the radius inside the circle inscribed inside the triangle.) Find an ...
0
votes
0answers
82 views

When is the “inequality” approach to limits valid?

For example, let's say $\lim_{x\to \infty} [g(x)]^{f(x)}=1$ . If we know that as $x \to \infty$, $h(x)> g(x)$ , we can say that $\lim_{x\to \infty} [h(x)]^{f(x)}$ equals $\infty$ . However, I ...
1
vote
1answer
58 views

Algebraic Divison

Is there a way to break the left hand side expression such that it takes the the right hand side form? $(a+b)/(c+d)=a/c+b/d+k$ Where $k$ is some expression.
5
votes
1answer
229 views

Sum of terms in a composition cycle

Let $f, g$ be linear functions. Define $S(x)$ as $any$ composition sequence of $f$ and $g$ like $S(x) = (f\circ g\circ g\circ f\circ f\circ g)(x)$ Let $s$ as the fixed point of $S$ then a cycle is ...
14
votes
2answers
371 views

how to compare $\sin(19^{2013}) $ and $\cos(19^{2013})$

how to compare $ \sin(19^{2013})$ and $\cos (19^{2013})$ or even find their value range with normal calculator? I can take $2\pi k= 19^{2013} \to \ln(k)= 2013 \ln(19)- \ln(2 \pi)=5925.32 \to k= ...
3
votes
2answers
609 views

Stuck on an 'advanced logarithm problem': $2 \log_2 x - \log_2 (x - \tfrac1 2) = \log_3 3$

I'm stuck on solving what my textbook calls an "advanced logarithm problem". Basically, it's a logarithmic equation with logarithms of different bases on either side. My exercise looks like this: $$2 ...
1
vote
3answers
234 views

How do I find the domain of a function?

Given the function $h(x)=\sqrt {5-x} $ and the function $f(x)=3x^2+\frac {6 } {x } -8$ how would I find the domain without graphing?
2
votes
1answer
1k views

Quadratic equation which has rational roots

If the following quadratic equation $$qx^2+(p+q)x+bp=0$$ always has rational roots for any non-zero integers $p$ and $q$ what will be the value of $b$? My book's solution says the value of ...
0
votes
1answer
12k views

Finding functions for an angle whose terminal side passes through x,y

How would I "Find the six trigonometric functions for the angle theta whose terminal side passes through the point (-8,-5)"?. I learned this material over 2 years ago and since then have forgotten. I ...
1
vote
1answer
1k views

finding the equation

A cell phone plan has a basic charge of \$35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost $C$ as a function of the ...
4
votes
1answer
209 views

How to find root of $ x^n+ax+b=0$?

I remember that there should be a formula for computing the root of $x^n+ax+b=0$. But I can't find it online. Could anybody point me the solutions? Thanks.
4
votes
2answers
281 views

Proving the inequality $\frac{a^3}{b^2-bc+c^2}+\frac{b^3}{a^2-ac+c^2}+\frac{c^3}{a^2-ab+b^2}\geq a+b+c$

I am trying to prove the following inequality For all positive numbers $a$, $b$ and $c$ we have $$\dfrac{a^3}{b^2-bc+c^2}+\dfrac{b^3}{a^2-ac+c^2}+\dfrac{c^3}{a^2-ab+b^2}\geq a+b+c$$ I can probably ...
1
vote
1answer
73 views

Help with graphing a piecewise function

What would be the graph and domain of this function? My domain is $(- \infty, \infty)$. I am stuck on graphing $-2x$. $$g(x)=\begin{cases} x+9 & \text{if }x<-3,\\ -2x & \text{if }|x|\leq ...
1
vote
1answer
86 views

domain of square root

What is the domain and range of square $\sqrt{3-t} - \sqrt{2+t}$? I consider the domain the two separate domain of each square root. My domain is $[-2,3]$. Is it right? Are there methods on how to ...
2
votes
1answer
81 views

Can you solve this problem with functions?

We are given $f(x)=ax^2+2x+b$, a is not $0$, $Df=R$ and $f\circ g=g\circ f$ where $g(x)=x$ has only solution $x_0$. Then we have to show that $ab\leq 1/4$.
2
votes
1answer
49 views

Equation with Side Conditions

A company has two different types of trucks: Type A has $20\,\text{m}^3$ space for goods which have to be cooled and $40\,\text{m}^3$ space for goods which do not have to be cooled. Type ...
0
votes
2answers
403 views

Work and Time related problem

A and B do a work together in 16 days. B and C can do the same work together in 20 days. How many days will it take for A, B and C to do the work together?
4
votes
3answers
220 views

How to prove that $\frac{1}{(1-x)^3}$ is the generating function for the triangular numbers?

How to prove that $\dfrac{1}{(1-x)^3}$ is the generating function for the triangular numbers? The $n^{\text{th}}$ triangular number is defined as $T_n = \displaystyle{n+1 \choose 2}$. I used ...