Tagged Questions

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
1answer
30 views

Best score in this puzzle

I want to maximise the score of the following table, choosing one item from each column/row, so no two items are on the same row or column. Score to maximise is just adding all the choices together. ...
1
vote
1answer
18 views

Equivalent of solutions of IVP

Consider the IVP $y''-2y'+26y=0$, $y(0)=1$, $y'(0)=2$. From the characteristic equation $m^2-2m+26=0$, i found the roots as $m_1=1-5i$ and $m_2=1+5i$. Then when i use the basis solutions ...
1
vote
0answers
19 views

Confused about the answer to the inverse of a cosine function

$$\arccos { (\cos { (\frac { 17\pi }{ 6 } ) } } )$$ No matter how I try and look at this problem, I end up with $\frac { 5\pi }{ 6 } $ I counted $\frac { \pi }{ 6 } $ 17 times counter clockwise ...
4
votes
4answers
89 views

Prove that $\sqrt{2} + \sqrt{3}$ is irrational. [duplicate]

Assume that $\sqrt{2}, \sqrt{3}, \sqrt{6}$ are all irrational. Prove that $\sqrt{2} + \sqrt{3}$ is irrational. So I know how to prove this using contradiction, and assuming that it is rational. But, I ...
0
votes
2answers
21 views

which rule did we use to simplify this logarithmic equation?

$$10^\frac{\log{x}}{log{10}} = x$$ I tried to use all the logarithm rules that i know but i still don't know how to derive this result, any hints please ?
2
votes
2answers
14 views

Lagrange interpolation for rational functions

Lagrange interpolation is very useful. I was wondering if there was an equivalent that is not using polynomials but rational functions, one polynomial divided by another. Look at this example: Say I ...
-4
votes
0answers
20 views

Find total amont of cookies? [on hold]

Find the total amount of cookies with this information: raspberry 20% vanilla 14% chocolate 48% banana 18%
0
votes
2answers
38 views

Multiplying a Rational Function by $-1$

I have the following examples when dealing with the results of multiplying by $-1$ and substituting with $-x$: $$f(x) = \frac{3x}{x^2-1}$$ $$f(-x) = \frac{-3x}{x^2-1}$$ $$-f(x) = ...
0
votes
2answers
28 views

solving for $x$, $x$ on both sides

I have $7(3x+6) = 11-(x+2)$. I read about the thing where you put one part of the equation on both sides after simplifying a bit to something like $21x+42 = 11-(x+2)$, using the distributive property, ...
3
votes
3answers
83 views

$\frac{1}{1+2}+\frac{1}{1+2+3}+\dots+\frac{1}{1+2+3+\dots+x}=\frac{2011}{2013}$

I want to see OTHER approaches than this one. Make sure they are significantly different and not a direct restatement. ...
1
vote
2answers
36 views

An equation for the third powers of the roots of a given quadradic polynomial

The roots of the equation $3x^2-4x+1=0$ are $\alpha$ and $\beta$. Find the equation with integer coefficients that has roots $\alpha^3$ and $\beta^3$. GIVEN SR: $\alpha + \beta = \frac43$ PR: ...
9
votes
3answers
56 views

Why does $n$-time differentiation of product have the same structure as raising sum to $n$th power?

A formula for differentiating a product is well known: $$(ab)'=a'b+ab'.$$ At first sight it doesn't resemble anything interesting. But what if we differentiate twice? We'll get ...
-3
votes
1answer
33 views

Four Complex Challenges [on hold]

re complex numbers such that $(x Given this, how many valu Suppose there are three complex numbers: $a$, $b$, and $c$ such that the three equations are true:, $b^2 + bc + c^2, - $c^ ca + ^2 = . ...
0
votes
2answers
28 views

Line tangent to the circle

Find the equation of the lines which are tangent to the circle with equation $x^2+y^2=9$ and parallel to the line of equation $x-y+1=0$
0
votes
0answers
18 views

Can I get the input from the output?

I have the formula for the center of gravity. And as an example of how I am using it; this is what it looks like when run (slightly modified) in javascript: ...
0
votes
1answer
8 views

Pre-Calculus Domain & Discontinuity

I'm doing some corrections on a test and this question has me stumped: David wants to create a rectangular holding area for his baby dragons, but only has 100 feet of fencing. He is going to use the ...
-3
votes
1answer
24 views

Math - problem solving [on hold]

If Superman can build a house by himself in 8 hours, and Batman & Robin together can build a house in 6 hours, and Robin & Superman together can build a house in 4 hours, how long would it ...
1
vote
0answers
30 views

2 point in time line

I have two times. For example, $t_1$= 0:17, $t_2 = 0:12$, and the time2WasBefore = 0:04, which means that $t_2$ was before now 0:4 minutes. I have to find the nearest situation, when $t_1$ will be one ...
3
votes
3answers
85 views

Why is $|x| \neq x$?

I was just thinking about the absolute value function. Why does the following equality not hold? $$|x| = \sqrt{x^2} = (x^{2})^{1/2} = x^1 = x$$ After all, there are clearly some values of $|x|$ that ...
1
vote
1answer
42 views

Natural Logarithm - solve the equation

I am having problems understanding how to solve $e^{4x}+4e^{2x}-21 = 0$.
3
votes
2answers
66 views

Roots of $2^{\frac x2} + (\sqrt{2} +1)^x = (5+2 \sqrt{2})^{\frac x2} $

I tried differentiating the equation to get minima and maxima, but failed to find the roots even there. Trial and error provides the answer=2, however, I'm searching for a proper method. Thanks in ...
3
votes
3answers
56 views

how to find the solution to this system of equations

Given the system of equations: $ xy+xz=54+x^2 \\ yx+yz=64+y^2 \\ zx+zy=70+z^2 $ Need to find all of the solutions of $ x,y$ and $z$. Tried to sum up all three equations but got stuck with ...
0
votes
1answer
28 views

invertible ring r -1

False the ring of integers: R=Z. The only invertible elements in this ring are 1 and −1. To make this clear, suppose a∈Z is invertible. then there is some a^-1∈Z such that aa^-1=a^-1a=1. But aa^-1=1 ...
3
votes
2answers
64 views

finding the positive integre solutions $x^2+5xy=5775$

Given the equation: $x^2+5xy=5775$. How can i find the positive integer solutions of it? I can write it as $x(x+5y)=5775$ but it seems to be hard to guess. Thanks.
0
votes
4answers
35 views

Trigonometric Question [on hold]

Show that: $$\frac{\cos \theta}{1+\sin \theta}+\frac{1+\sin \theta}{\cos \theta} = \frac{2}{\cos \theta}$$
2
votes
3answers
28 views

Conjectured Inequality

I noticed that for positive integers $(a,b,c)$, it happens that $ab+bc+ca$ is always greater than $abc+1$. Is this true, and if it is, how would I prove it?
-2
votes
1answer
18 views

Composition of Continuous and discontinuous function [on hold]

Please help me to find an example of each composition of f and g is continuous Composition of f and g is discontinuous whereas f is a continuous function and g is discontinuous function
1
vote
3answers
41 views

How do you solve a system of equation with a variable as an exponent?

So, I've been doing some random math on my own and I ran into a problem. I was met with the system of equations of $$x=3^y$$ $$x+1=5^y$$ How can I solve this algebraically without using the graphs?
3
votes
1answer
30 views

Value of sum $\sum_{i=0}^n\binom{2n}{2i}(-3)^i$

What is the value of the sum $\sum_{i=0}^n\binom{2n}{2i}(-3)^i$? It looks like the binomial expansion $(1+x)^n=\sum_{i=0}^n\binom{n}{i}x^i$, but we only take every other term, and also the power is ...
0
votes
2answers
27 views

Please help me solve this (f+g)(-2) when f(x)= x-3 and g(x)=x+1?

Can anyone help me solve this or please post a step by step text on how to get this problem set up. I am not understanding why I cannot get the right answer for this one. It should be -6 but I am not ...
1
vote
1answer
20 views

I don't understand this step in this exercise involving algebra

In the solution of an exercise they go from $e^{2y} = x + x\cdot e^{2y}$ straight to $e^{2y} = x / (1-x)$. Could anybody explain to me how/why this is possible?
1
vote
0answers
29 views

Isoperimetrics question

Here's an Algebra 2 problem that has been giving me fits. The question is: A 50-inch length of string is cut and one length is formed into a circle and the other into a square. Where should you cut ...
3
votes
0answers
26 views

Condition $|x_1x_2+1|<x_1+x_2$ in quadratic polynomial

Let $x^2-ax+b$ be a polynomial with real coefficients having two nonzero roots. Given that $|b+1|<a$, and one of the roots have modulus $<1$, show that the other root has modulus $>1$. We ...
2
votes
0answers
66 views

Isolating x and z in two equations.

I am working on a computer program and at some point I need to isolate an x and a z. I am basically trying to isolate x and z in these two equations: 1) $xn_{x} + yn_{y} + zn_{z} = n_{d}$ 2) ...
0
votes
0answers
18 views

Arithmetic Reasoning- what time will the man take to do the job alone?

A man and woman can do a job in 60 days. If the ratio of their work is $13:7$, what time will the man take to do the job alone?
1
vote
3answers
45 views

How to calculate with respect to money??

if A and B undertake to do a piece of work for Rs.108 . A can do it in 5 days while B can do in 7 days . if both of them finished the work together , how much shell B get s? I am confusing with ...
-4
votes
0answers
24 views

Need Help? Need answer sheet to check answers! Not test [on hold]

I have a test coming up, and Our entire class was given a review sheet to work on. I want to know if my answers are correct, and my teacher is out of school this week. I have a digital PDFs copy Which ...
2
votes
1answer
72 views

How to show $\frac {\cos(x)+\sin(x)}{\cos(x)-\sin(x)}=\frac{1+\tan(x)}{1-\tan(x)}$

A step in trig expression simplification, from a textbook: $$\frac {\cos(x)+\sin(x)}{\cos(x)-\sin(x)}\to\frac{1+\tan(x)}{1-\tan(x)}$$ Please give a hint on how was this transformation achieved.
-3
votes
0answers
25 views

Writing Expressions [on hold]

a “The program is faulty”, s “I will lose my data” d “The disc is full”. Write an expression in terms of a, s and d that exactly represents the proposition “In the case of a faulty program, I will ...
1
vote
0answers
21 views

Basic Algerbra - Ratio between three numbers.

I'm doing an analysis, of client contracts that contain the following factors: The per-unit license price (A) The initial order quantity (B) The recommended retail price (C) What I'd like is a ...
3
votes
1answer
23 views

Reason behind solution in this inequality with absolute values

Solve the inequality $|3x-2|-|x+2|>x$ When $|x+2|<0$: $-(3x-2)+(x+2)>x\iff x <\frac{4}{3}$ When $|x+2|>0\land |3x-2|<0$: $-(3x-2)-(x+2)>x\iff x < 0$ When $|3x-2|>0$: ...
-1
votes
1answer
19 views

MathWordProblem : Time and Work

A can do a job as fast as B and C working together . If B does the job alone in 5 days and C doest it in 6 days . how many day s does it take A to to a job alone
0
votes
3answers
14 views

Slope from a graph

I was wondering if I have a graph with no points specified and they only say find the slope how do I do it like what am I supposed to do and how can I find out the points
0
votes
1answer
22 views

What is the remainder when dividend is divided by product of divisor with given remainder?

When $ P(z) $ is divided by $ z+1 $ the remainder is $ -8 $ when $ P(z) $ is divided by $ z-3 $ the remainder is $ 4 $ find the remainder when $ P(z) $ is divided by $ (z-3)(z+1) $?
4
votes
1answer
76 views

How to find common ratio of a geometric series.

I have this series: $$S_n = \sum_{k=1}^\infty \frac{5}{k} = 5+\frac{5}{2} + \frac{5}{3} + .... + \frac{5}{k}$$ I wonder if theres a method to find the $r$ for any geometric series, Because sometimes ...
0
votes
3answers
43 views

Show that $a \lt \frac{a + b}{2} \lt b$ for $a\lt b $ and $a, b \in \mathbb{R}$

How can I prove this statement true? I have tried saying starting like this: $a = 0; \qquad b>0.$ But I don't know where to proceed from here
0
votes
2answers
20 views

Finding vertical asymptote

When finding the vertical asymptote of a rational function why can only the denominator$ = 0$ and not both the numerator and the denominator? both are undefined for $x$... For example, for $F(x) = ...
0
votes
2answers
65 views

What does 8 root 2.5 ($\sqrt[8]{2.5}$) mean?

$8$ root of $2.5$? I have calculated the equation and found $12.6$ is the answer. However, the answer given is $1.12$ so what could I have done wrong? Thanks
6
votes
1answer
70 views

Solve $3x^2-y^2=2$ for Integers

If $x$ and $y$ are integers, then solve (using elementary methods) $$3x^2-y^2=2$$ I tried the following If $y$ is even, then $4|y^2$ and hence $2|y^2+2$ (and $4$ doesn't divide it), but ...
3
votes
2answers
47 views

prove that $a^3 - b(b+1) = (a-2)(c^2 + 1) + 2$ has infinitely many solutions

How do I prove that $$a^3 - b(b+1) = (a-2)(c^2 + 1) + 2$$ has infinitely many solutions if a, b and c are natural numbers? I have opening the brackets and moving all the terms to one side which gets ...