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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0
votes
1answer
12 views

no. of distinct real roots of the equation $x^2=x\sin x+\cos x$

$(1)$ The no. of Distinct real solution of the equation $x^4-4x^3+12x^2+x-1=0$ $(2)$ The no. of distinct real roots of the equation $x^2=x\sin x+\cos x$. $\bf{My\; Try}$ For $(1)$ one:: Let ...
1
vote
2answers
71 views

Evaluation of $\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\sum_{k=0}^{\infty}\sum_{l=0}^{\infty}\frac{1}{3^{i+j+k+l}}\;\;,$ Where $i\neq j \neq k\neq l$

Evaluation of following Infinite Geometric series. $(a)\;\; \displaystyle \sum_{i=0}^{\infty}\sum_{j=0}^{\infty}\frac{1}{3^{i+j}}\;\;,$ Where $i\neq j\;\;\;\;\;\;\;\;\;\; (b)\;\; \displaystyle ...
-4
votes
3answers
17 views

Find the gradient of a line [on hold]

If the $x$-intercept of a line is $3$ and the $y$-intercept is $-2$, what is the slope of the line?
-7
votes
1answer
24 views

Points, Slopes and Intercepts [on hold]

Find the value of $X$ so that the line passing through the two points has the given slope: $(X,-7)$ and $(-2,9); M=-4$ If the $X$-intercept of a line is $3$ and the $y$-intercept is $-2$, what is the ...
0
votes
1answer
26 views

Need a step by step on finding the average rate of change for $x^3-2x+8$

I am trying to solve a rate of change problem. The answer is given but I do not understand where the math comes from. I am new to this subject in algebra. It is asking for the average rate of change ...
2
votes
1answer
28 views

Show that the permutation [n, n-1,…, 2,1] has n(n-1) inversions

Show that the permutation $[n, n-1,..., 2,1]$ has $n(n-1)$ inversions How do I show that this is true? Why isn't $(n(n-1))/2$
-1
votes
0answers
39 views

How can I do better in algebra 1? [on hold]

I get 60s in every exam. I have 2 weeks left in the quarter. How can I do better? I understand the math. I get all the homework and classwork, but when the exams come, I do very bad. What can I do to ...
0
votes
1answer
23 views

Intersection between line and curve?

I want to find intersection between line(y=x) and a curve for instance y=tanh(x) I know they intersect on some points I don't know algebraically how to find that for example I tried $tanh(x) = x$ ...
0
votes
2answers
29 views

how to calculate speed of two object that are moving toward each other?

Two cars started their journey from point A and B 150 km apart on the same road towards each other.The car started from A traveled at a constant speed 10 km/hr more than that of hat also traveled at a ...
1
vote
1answer
45 views

Chemistry or Mathematics

Suppose that I have two mixtures, V and W. I make a solution A by mixing V and W in a 2:5 ratio. I make another solution B by mixing V and W in a 3:4 ratio. I then make one more solution C by mixing A ...
1
vote
0answers
22 views

what is the logic to solve this question ?

If the value of a particular stock increased by 10% every day of the first four days of a week.However, it value decreased by 30% at the end of fifth day compared to its value at the end of the fourth ...
0
votes
3answers
33 views

Prove $(x^2-2)\sin(x)+2x\cos(x)\geq 0$ for $x$ on $[0,\pi/2)$

Originally comes from the question How to prove SinA/A+sinB/B+SinC/C<(9*(3)^.5)/2pi. The goal is proving $\frac{\sin(x)}{x}$ concave down on $[0,\pi/2)$, which I find it non-trivial.
1
vote
3answers
26 views

Tricky Decomposition

Decompose the following polynomial: $A=x^4(y^2+z^2) + y^4(z^2+x^2) + z^4(x^2+y^2) +2x^2 y^2 z^2 $ (by the way I'm not quite sure about the tags and the title feel free to edit them if you wish)
0
votes
3answers
28 views

Basic algebra/fractions: derivation

How do I do this derivation step? I don't understand why there is equality. The derivation is from my textbook. $$mg-m\left(\frac{g}{1+\frac{M}{2m}}\right)=\frac{mg}{1+\frac{2m}{M}}$$
0
votes
1answer
48 views

How to prove SinA/A+sinB/B+SinC/C<(9*(3)^.5)/2pi

Only for an acute angle triangle. $A$,$B$,$C$ are angles of a triangle. This isnt sine rule form. Ive tried Cauchy Schwarz theorem , A.M, G.M form but am unable to get the above result. Could someone ...
1
vote
1answer
23 views

how to transform word problem to an equation ?? is there any trick for this?

A team f eleven spies were assigned consecutive whole numbers divisible by eleven as their identity code . For telephonic contact they have to use contact code that equals the product of the middle of ...
0
votes
4answers
16 views

how to proceed ? Have I correctly transformed the question to an equation?

Amongst Teaching staff of ABC university the ratio of men and women is $5:2$. Amongst the women $\large\frac{3}{7}$ are not married . If the number of married women teacher is $56$ then the total ...
7
votes
3answers
88 views

Inequality $(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$

Let $n\geq 2$, and $x_1,x_2,\ldots,x_n>0$. Show that $$(1+x_1)(1+x_2)\ldots(1+x_n)\left(\dfrac{1}{x_1}+\dfrac{1}{x_2}+\cdots+\dfrac{1}{x_n}\right)\geq 2n^2.$$ For $n=2$, this reduces to ...
2
votes
1answer
38 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
21 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
1answer
24 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
99 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
24 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
19 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
40 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
29 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
85 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
57 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
34 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
19 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
88 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
43 views

Natural Logarithm - solve the equation

I am having problems understanding how to solve $e^{4x}+4e^{2x}-21 = 0$.
3
votes
2answers
73 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
58 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
65 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
31 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?
4
votes
1answer
31 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 ...