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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1
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3answers
90 views
+50

Find the latus rectum of the Parabola

Let $y=3x-8$ be the equation of tangent at the point $(7,13)$ lying on a parabola, whose focus is at $(-1,-1)$. Evaluate the length of the latus rectum of the parabola. I got this question in ...
1
vote
1answer
20 views

What is the coefficient and constant term in the following sequence defined recursively?

Let $f_n(x)$ be a sequence of polynomials defined inductively as $f_1(x) = (x - 2)^2$ $f_{n+1}(x) = (f_n(x) - 2)^2$ $; n \ge 1$ Let $a_n$ and $b_n$ respectively denote the constant term and the ...
0
votes
1answer
34 views

What is the way to solve the following system of equations?

What are the methods of solving the following system: $x+y+z=4$ $x^2+y^2+z^2=24$ $x^3+y^3+z^3=64$
2
votes
2answers
77 views

We have $a^3+b^3$ and $ab$, how we can calculate $a+b$?

One of my friends is a high school student, he asked me this question. It's soluble by use of General formula for cubic roots, because: $$(a+b)^3=a^3+b^3+3ab(a+b)$$ But he looked for a simple ...
2
votes
2answers
31 views

Time-and-Work and Motorcycle Tyres

A problem about motorcycle tyres, related to Time-and-Work or rate-of-work methods. This is not a homework question, nor, as far as I know, a contest question. It is intended as a challenge for Year ...
-5
votes
1answer
18 views

Finding the equation for an ellipse in $y$ form [on hold]

given $\frac{x^2}{2500} + \frac{y^2}{2500} = 1$ how do I solve for the $y=$ form of the equation? I assume there will be a $+/-$ of the same equation but have no idea how to solve
7
votes
2answers
37 views

Show that the curve $\dfrac{x^2}{a^2} +\dfrac{ y^2}{b^2} = 1$ form an ellipse

If the definition of an ellipse is the set of points $(x,y)$ such that given two focus points $F_1, F_2$ the sum of the distances from $(x,y)$ to each focus point is constant, how can one show that ...
2
votes
3answers
39 views

Precalculus/Trigonometric Functions of Sine, Cosine, and Tangent with given parameters?

for my precalculus class I was given an assignment for extra credit however it is some material that I have yet to cover or learn as far as sine, cosine, and tangent go. Below is the prompt that I was ...
1
vote
1answer
24 views

Value of $a$ such that range contains the interval $[0,1]$

Find the number of integral values of $a$ in the interval $[0,100]$ so that the range of the function $y= \frac{x+a}{x^2-1}$, $x\in R$ contains the interval $[0,1]$? After rearranging $y= ...
4
votes
4answers
504 views

How to solve certain types of integrals

I'm asking for a walk through of integrals in the form: $$\int \frac{a(x)}{b(x)}\,dx$$ where both $a(x)$ and $b(x)$ are polynomials in their lowest terms. For instance $$\int ...
19
votes
4answers
382 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+1/3 }{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
0
votes
1answer
18 views

Whats the age of Bill's Kids?

I was going through some problems.This one i didn't get the solution can anyone please help me in solving this. Two old friends, Jack and Bill, meet after a long time. Jack: Hey, how are you man? ...
0
votes
1answer
76 views

How to solve $2^x < x^2$

How do you solve : $$2^x < x^2$$ My math years are behind me, so I can't wrap my head around how to continue after this step : $$2^x - x^2 < 0$$ I think there's a trick since it's a 0 ...
0
votes
1answer
467 views

Irrational inequalities question: $\sqrt { -3x+1 } + \sqrt {6x+1} < \sqrt {3x+4}$ and $\sqrt { -6x+10 } + \sqrt {-x+2} \gt \sqrt {4x+5}$

Consider the following inequalities: \begin{equation*} \sqrt { -3x+1 } + \sqrt {6x+1} \lt \sqrt {3x+4}, \\ \sqrt { -6x+10 } + \sqrt {-x+2} \gt \sqrt {4x+5}. \end{equation*} Attempt at a solution; ...
1
vote
0answers
21 views

Number of integral solutions of $x_1.x_2.x_3=x$

Let $x$ be the element of the set $A=\{1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120\}$ and $x_1, x_2, x_3$ be positive integers and $d$ be number of integral solutions of $x_1.x_2.x_3=x$ , then $d$ is ...
3
votes
2answers
55 views

simple proof for principle of pigeons

I must prove the principle of pigeons but the proofs I find in the internet are too complex. Here's what I can use: Definition $$I_n = \{p\in \mathbb{N}; p\le n\}$$ The principle of the pigeons ...
0
votes
1answer
78 views

Logarithm in the exponent

$$(2x)^{\log 2} = (3y)^{\log 3} \\ 3^{\log x} = 2^{\log y}$$ Solve for $x$ and $y$. My intuition for solving such problems is taking the logarithm on both sides but it does not work. I also ...
2
votes
1answer
61 views

A simple cubic equation problem:

Consider the cubic equation $$az^3-bz^2+\bar{b}z-\bar{a}=0$$ where $a$ and $b$ are non-zero complex numbers. Suppose $z_1, z_2$ and $z_3$ are the roots. Question: Which $a$ and $b$ gives ...
3
votes
2answers
78 views

Evaluating $\int_{0}^{3} \sqrt{1+x}\: dx$ using Limit of a Sum approach

Evaluate $\int_{0}^{3} \sqrt{1+x}\: dx$ using Limit of a Sum approach. Using the formula $$\int_{a}^{b} f(x)\:dx=(b-a) \times \lim_{n \to \infty} \frac{1}{n} \times ...
1
vote
1answer
71 views

Find all solutions of the equation $3 \cdot 2^{x+2}+5^x=8\cdot 3^x+5$

Find all solutions of the equation $$3 \cdot 2^{x+2}+5^x=8\cdot 3^x+5$$ My work so far: Let $f(x)=3 \cdot 2^{x+2}+5^x-8\cdot 3^x-5$ $f'(x)=12\cdot 2^x\ln2+5^x\ln5-8\cdot3^x\ln3$
3
votes
2answers
35 views

$\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$

I must prove: $\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$ Well, I know that $$\lim x_n = a \implies |x_n-a|<\epsilon$$ $$\lim \frac{x_n}{y_n} = b \implies ...
1
vote
1answer
22 views

$a_n, b_n$ bounded, $a_n+b_n=1$,$z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$

I must show that if $a_n, b_n$ are bounded such that $a_n+b_n=1$, and if $z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$ My idea was: $$(a_n+b_n)(z_n+t_n) = a_nz_n+a_nt_n+b_nz_n+b_nt_n$$ I ...
0
votes
1answer
35 views

Is this function increasing/decreasing and convex/concave?

The function is: $$y= 3x + \ln\left(\frac {3x - 4}{x - 1}\right)$$ After differentiating I got: $$y' = 3 + \frac 1{3x^2 - 7x + 4}\;\;\;\;\; \;\;\; y''= - \frac {6x - 7}{(3x^2 - 7x + 4)^2}$$ ...
0
votes
0answers
11 views

little agebra help, complex numbers

Can someone please explain this to me, I dont understand how to go from $ [ \psi-1+r( e^{2i\omega} - 4e^{-\omega}+6-4e^{-i\omega} + e^{-2i\omega})] A\psi6{n}e^{i\omega j} $ to this line here. $ ...
3
votes
6answers
118 views

Prove: $\frac{a+c}{b+d}$ lies between $\frac{a}{b}$ and $\frac{c}{d}$ (for positive $a$, $b$, $c$, $d$)

I am looking for proof that, if you take any two different fractions and add the numerators together then the denominators together, the answer will always be a fraction that lies between the two ...
1
vote
2answers
38 views

A human way to simplify $ \frac{((\sqrt{a^2 - 1} - a)^2 - 1)^2}{(\sqrt{a^2 - 1} - a)^22 \sqrt{a^2 - 1}} - 2 a $

I end up with simplifying the following fraction when I tried to calculate an integral(*) with the residue theory in complex analysis: $$ \frac{((\sqrt{a^2 - 1} - a)^2 - 1)^2}{(\sqrt{a^2 - 1} - a)^22 ...
2
votes
7answers
87 views

Why does the power series of $ x + x^2 + x^3 …$ not equal to $x/(1-x) $ when x is larger than 1?

Why does the power series of $x + x^2 + x^3\ldots$ not equal to $x/(1-x)$ when $x$ is larger than $1$? It is never specified what range of values $x$ takes on, so the algebra should work out in all ...
1
vote
1answer
59 views

how do I solve $y -\sin y= 1$

I am trying to use trigonometric equations to solve $y - \sin y = 1$, such as solving for $y$ but it is not working out, I have found $\cos y = \sqrt{-y^2 + y}$ but it does not lead to anywhere ...
0
votes
2answers
24 views

Multiplying binomials to come up with $ y^8 - 256 $

$$\ { (y^4 + 16) }{ (y^2 + 4) }{ (y + 2) }{ (y - 2) }$$ How do I multiply these to come up with $\ {y^8 - 256}$
0
votes
1answer
25 views

What is this vector equation?

I was going through some documents and I came across this vector equation (the vector is composed of a real part and imaginary part): if: $ v = a + j*b $ then: $w = \sqrt(|a|) * \sqrt(\frac{1 + ...
-3
votes
3answers
54 views

Can someone please explain how $60+\ln(64)-\ln(8)$ is equal to $60+\ln(8)$ [on hold]

Can someone please explain how $60+ \ln(64)- \ln(8)$ is equal to $60+\ln(8)$. I can't understand why this is true.
1
vote
1answer
28 views

Solve $\log_9(x-4) - \log_9(x-8)= \frac{1}2$

Solve $\log_9(x-4) - \log_9(x-8)= \frac{1}2$ $(x-4) - (x-8)= 9^\frac{1}2$ $(x-4) - (x-8)= 3$ The answer is 10 but I am not sure how that was obtained.
1
vote
1answer
23 views

How many ways are there to arrange the letters of word $ALGEBRA$ such that the relative order of the vowels and consonants doesn't change?

I did this question this way :- there are 4 consonants in the words (LGBR) and there are 7 letters in the word. $therefore$ number of in which consonants can be arranged in relative order will be ...
0
votes
2answers
54 views

How to solve $y^2 \ge x^2$?

How can I decide where to write the minus after taking the square root? Could someone explain this? $y^2 \ge x^2$
-1
votes
0answers
7 views

Gas Turbine Performance and square root functions [on hold]

The formula for verifying a variable vane angle as related to a given compressor speed is; corrected speed (SpdK) = speed(1√Θ). I assume that Θ = observed angle at a given speed. If √Θ is the same ...
-2
votes
3answers
72k views

What is the formula for the difference between CI and SI?

if principal, time and rate are given how do i find the difference between Compound interest and Simple Interest? P=12,000 n=1 and a 1/2 yrs. R=10% per year ...
0
votes
1answer
981 views

What will be the multiplicative inverse of square root of 5 with respect to a natural number $M$?

Can such a number $N$ be found such that $\sqrt{5}N \equiv 1 \mod M$? If no,what can be the best approximation for $N$?
3
votes
4answers
118 views

Quick way to solve the system $\displaystyle \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} = \frac{65}{36}$, $xy-x+y=118$.

Consider the system $$\begin{aligned} \left( \frac{3}{2} \right)^{x-y} - \left( \frac{2}{3} \right)^{x-y} & = \frac{65}{36}, \\ xy -x +y & = 118. \end{aligned}$$ I have solved it by ...
2
votes
3answers
38 views

$f(x)=e^x -10x^2$ doesn't vanish in more than three points.

How can I prove that $f(x) = e^x - 10x^2$ doesn't vanish in more than three points. I stuck here I just computed the derivative that is $f'(x) = e^x - 20x$ and then $x= \log(20)+\log(x)$ when ...
4
votes
0answers
194 views
+500

Bounding a sum involving a $\Re((z\zeta)^N)$ term

This is a follow up to this question. Any help would be very much appreciated. Let $k\in\mathbb{N}$ be odd and $N\in\mathbb{N}$. You may assume that $N>k^2/4$ or some other $N>ak^2$. Let ...
1
vote
1answer
29 views

What is the determinant of cofactor matrix of a matrix? [duplicate]

For an $n \times n$ square matrix $A$, can determinant of its cofactor matrix (matrix consisting of cofactors of the elements of $A$) be expressed in terms of $\det(A)$ and $n$ ?
-3
votes
1answer
22 views

Time Speed and Distance(train) [on hold]

A train crosses a man travelling in another train in the opposite direction in 8 seconds.However,the train requires 25 seconds to cross the same man if the trains are travelling in the same ...
0
votes
1answer
14 views

Trouble while dividing a product

First of all sorry for my bad english. I stumbled upon this term. Is it possible to transform it like that? $$F\times r_1=\frac Q 2\times (r_1-r_2)$$ $$F=\frac Q 2\times \frac {(r_1-r_2)}{r_1}$$ ...
91
votes
23answers
12k views

If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself?

If squaring a number means multiplying that number with itself then shouldn't taking square root of a number mean to divide a number by itself? For example the square of $2$ is $2^2=2 \cdot 2=4 $ . ...
707
votes
25answers
113k views

How long will it take Marie to saw another board into 3 pieces?

So this is supposed to be really simple, and it's taken from the following picture: Text-only: It took Marie $10$ minutes to saw a board into $2$ pieces. If she works just as fast, how long ...
1
vote
1answer
18 views

Expressing the minimum function in terms of the absolute value in a symmetric manner (generalized to more variables)

It is well known that: $$\max(a,b) = \frac12(a+b)+\frac12|a-b|$$ and similarly: $$\min(a,b) = \frac12(a+b)-\frac12|a-b|.$$ In fact, they are equivalent since $\max(a,b) = -\min(-a,-b)$. We can try ...
1
vote
0answers
31 views

Possible to isolate x for: y = x + sin(x)

I just recently learned that: $x = \frac{b}{a + 1}$ when $ ax + x = b$. This got me thinking about trig functions in a similar format, i.e. can we isolate $x$ for: $y = x + sin(x)$?
4
votes
2answers
50 views

What does $\Big(\frac{(x+1)^2}{2}\Big)^n-\Big(\frac{(x-1)^2}{2}\Big)^n$ equal to?

Determine the highest degree term of the polynomial $$\Bigg(\frac{(x+1)^2}{2}\Bigg)^n-\Bigg(\frac{(x-1)^2}{2}\Bigg)^n, \quad n\in\mathbb{N}$$ The answer suggests that the highest degree term is ...
1
vote
2answers
69 views

How do I solve for $m$ and $n$

While reading about nested radicals, I came across a theorem that said $\sqrt{m\sqrt[3]{4m-8n}+n\sqrt[3]{4m+n}}=\pm\frac ...
1
vote
1answer
107 views

How to solve $x^3-3x= \sqrt 3$

$$x^3-3x= \sqrt 3$$ I have tried solving above equation using trial error method with many alternatives. I reached just near in decimal number. Actually, I need the radical exact notation, which will ...