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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0
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2answers
68 views

Changing from x to y $x = y(4-y)$

$$x = y(4-y)$$ I am guessing I need some pretty advanced math to solve this for y. I am trying to use the shell method and I have to use opposite terms of the rotation axis so I am rotating around y ...
1
vote
2answers
623 views

Find the derivative of $y = f(x^2 - 2x + 7)$ where $f'(10) = 2$

Determine the derivative if $y = f(x^2 - 2x + 7)$ and $f'(10) = 2$ Ok so honestly, I dont know how to solve this, or even know where to start. All i know is that we are given a point $(10, 2)$. But ...
0
votes
0answers
106 views

Simplify the math expression $M=(1-b)^{N}+(N-4)(1-b)^{N-1} b+ \sum^{N}_{i=1} 2^{2i-2} \Delta^2 (1-b)^{N-1}b-(1-2b)^2$

Can someone help me to further simplify the following expression? Here, $0<b<1$ and we can assume that $b$ is small. $\Delta$ is a constant. Thank you $M=(1-b)^{N}+(N-4)(1-b)^{N-1} b+ ...
4
votes
3answers
150 views

Why $\sum_{k=1}^n \frac{1}{2k+1}$ is not an integer?

Let $S=\sum_{k=1}^n \frac{1}{2k+1}$, how can we prove with elementary math reasoning that $S$ is not an integer? Can somebody help?
0
votes
2answers
232 views

Mix-problem with percentage

A can is containing coffee and another can is containing exactly the same amount of milk. We take a spoon of coffee and mix it in the milk can. Then we take a spoon from the mix we obtained and mix it ...
-1
votes
1answer
60 views

State the equation of the line with $x$-intercept $x=5$ and $y$-intercept $y=-2$

I really don't know how to even state the equation. I only know how to find out the $x$ and $y$ intercepts when I know the equation of the line. However, the options are: a. $ 2x - 5y + 10 = 0$ b. ...
1
vote
2answers
128 views

Determining an expression problem

Determine an expression, in simplified form, for the slope of the secant $PQ$ with $P(1,2)$ and $Q(1+h, f(1+h))$ where $f(x) = 2x^2$ I don't even know how to approach such a question. Any help ...
-1
votes
2answers
116 views

Which of the following is not used to determine the slope of a function algebraically?

I dont know the answer to the above question. I think it is the slope of a secant line but I'm not sure ...
0
votes
1answer
108 views

Norm inequality (supper bound)

Do you think this inequality is correct? I try to prove it, but I cannot. Please hep me. Assume that $\|X\| < \|Y\|$, where $\|X\|, \|Y\|\in (0,1)$ and $\|Z\| \gg \|X\|,\|Z\| \gg \|Y||$. prove that ...
3
votes
3answers
784 views

The Product Rule of Square Roots with Negative Numbers

In the statement $\forall a, b \geq0, \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$, why is it necessary to restrict $a$ and $b$ to being $\geq 0$? It seems that one should be able to say, for example, ...
4
votes
3answers
179 views

Why don't we define division by zero as an arbritrary constant such as $j$? [duplicate]

Why don't we define $\frac 10$ as $j$ , $\frac 20$ as $2j$ , and so on? I know that by following the rules of math this eventually leads to $1=2$ , but we could make an exception and say that $j$ is ...
1
vote
1answer
114 views

Trigonometric manipulation of complex number, how does this step occur?

I was reading the section about DeMoivre, and my book showed how to derive his formulas. The next part is supposed to be about finding roots of complex and real numbers. Roughly, it says: "Let $z$ be ...
0
votes
1answer
58 views

Find the integer solutions

What are the pairs $(A,N)$ where $A,N$ are integers such that the following equation is satisfied: $\large A=\frac{-6+\sqrt{144-12N^2}}{6}$ I know that we should have: $k^2=144-12N^2$ for some ...
3
votes
1answer
434 views

Find $\cos(2\alpha)$ given $\cos(\theta -\alpha)$ and $\sin(\theta +\alpha)$

My question is: If $\cos(\theta -\alpha) = \frac{3}{5}$ and $\sin(\theta +\alpha) =\frac{12}{13}$, find $\cos(2\alpha)$. Attempt I: \begin{align*} &\cos^2(\theta -\alpha)+\sin^2(\theta ...
0
votes
1answer
136 views

Optimizing $x^2+y^2$ from two given equations? [duplicate]

What is the maximum value of $x^2+y^2$, where $(x,y)$ are solutions to: $$2x^2+5xy+3y^2=2$$ and $$6x^2+8xy+4y^2=3$$ Note: Calculus is not allowed. I tried everything I could but whenever I got for ...
4
votes
2answers
352 views

$4$ women and $2$ men are being interviewed. Find the probability the women will be interviewed first.

My Calculations: $$\frac{4}{6}\times\frac{3}{5}\times\frac{2}{4}\times\frac{1}{3} = \frac 1 {15}$$ Is that correct?
3
votes
1answer
65 views

Given that $a^2(a+k)=b^2(b+k)=c^2(c+k)$, find the value of $1/a+1/b+1/c$

Given $$a^2(a+k)=b^2(b+k)=c^2(c+k)$$ find the value of $1/a+1/b+1/c$. I tried to derive a relation from the equality but it did not help my cause.
5
votes
2answers
178 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
38 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 ...
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
59 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
666 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
43 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
768 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
137 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
59 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
113 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
104 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
57 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
81 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?
1
vote
1answer
82 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
140 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
146 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
49 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
88 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
231 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
120 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
117 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
357 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
86 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.