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.

learn more… | top users | synonyms (2)

0
votes
0answers
5 views

Weird problem my friend told me.

So, this one's tricky. There's a keycode combination, and there are six buttons. Each button multiplies the base number of 1 by their respective multipliers (see below). Once the result number gets ...
1
vote
3answers
49 views

Why would the cubic have $5$ roots?

The polynomial $P(x)$ is cubic. What is the largest value of $k$ for which the polynomials $Q_{1}(x) = x^{2}+(k-29)x-k$ and $Q_{2}(x) = 2x^{2}+(2k-43)x+k$ are both factors of $P(x)$? $P(x) = ...
4
votes
0answers
51 views

Find the sum of the series below

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ What I have tried... We are looking for ...
3
votes
2answers
32 views

Find the least $N$ so there is no square

Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer. Let $x^2$ appear before $1000N$ so: $(x+1)^2 ...
1
vote
0answers
17 views

simplifying complex expression

Hi I am trying to simplify the following expression:$$ \left|\frac{1}{a+ib}\left(\frac{J_1(c x)}{J_1(c b)}-x\right)\right|^2,\quad a,b,x\in \mathbb{R}, \ c\in \mathbb{C} $$ Is there a simple way of ...
0
votes
0answers
20 views

Proof: Condition that two quadratic functions may have a common linear factor. [on hold]

Find the condition that two quadratic functions of $(x,y)$ called $ax^2 + bxy + cy^2 $and $a'x^2 + b'xy + c'y^2$ may have a common linear factor.
1
vote
3answers
26 views

Slope of a line segment.

If $A(x_1, y_1)$ and $B(x_2, y_2)$, we know that slope $m = \frac {(y_2 - y_1)} {(x_2 - x_1)}$. What decision can we take aout the line segment when, $m = \frac 0 0$, $m = \frac {dy} 0$, and, $m = ...
0
votes
2answers
48 views

How many divisors of the combination of numbers?

Find the number of positive integers that are divisors of at least one of $A=10^{10}, B=15^7, C=18^{11}$ Instead of the PIE formula, I would like to use intuition. $10^{10}$ has $121$ divisors, ...
1
vote
1answer
61 views

Solving $\frac{9a^3-7ab^2+2b^3}{3a+2b}=3a^2-2ab-b^2+\frac{4b^3}{3a+2b}$

I have the following problem: $$\frac{9a^3-7ab^2+2b^3}{3a+2b}$$ The solution in the book is $$3a^2-2ab-b^2+\frac{4b^3}{3a+2b}$$ but I do not know how to get there. I could solve the other ...
2
votes
2answers
22 views

Show that any 2D vectors can be expressed in the form…

(a) Show that any 2D vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let $u$ and $v$ be ...
1
vote
1answer
48 views

Two infinite radicals question

Hello I have stucked with theese two questions: $\sqrt{a:\sqrt{a:\sqrt{a: \cdots}}} + \sqrt[3]{a\cdot\sqrt[3]{a\cdot\sqrt[3]{a\cdots}}} = 12$ $a=\text{ ?}$ ...
3
votes
4answers
30 views

Manipulating equations question

In the equation: $$T = 2\pi \sqrt {\frac lg}$$ it is for determining period of pendulum swing If I want to solve for $g$ and I want to start by removing the root do I square everything in the ...
1
vote
1answer
35 views

How to use Principle of Inclusion-Exclusion here?

A while ago I posted a question: Coloring a Grid. Online, I seem to have stumbled upon a usage of PIE AOPS Wiki Solution AIME II #9. (1) Now, I have experience with PIE, but I do not see how to ...
0
votes
4answers
81 views

$2n=n^2$ what are the solutions

I have just thought of this and I know someone must have before but is the only solution (with real numbers) to $2n=n^2$ $n=2$
1
vote
1answer
30 views

Why is $2^4$ congruent to $-1$ modulo $17$?

I saw an interesting question on Quora (What remainder is obtained when $2^{2017}+1$ is divided by $17$?), but I do not understand the author's solution: Three, because $$ \begin{align} 2^{2017} ...
0
votes
1answer
24 views

Converting complex query to algebra

Algebra is all I know, and I cannot resolve this using my rudimentary algebra. What is the correct procedure? On Wednesday all items at a clothing store were $15. Brenda bought a number of ...
-1
votes
1answer
22 views

Algebra with two variables [on hold]

I have tried answering the following question using algebra, but as it has two variables, I don't know how to solve. Please advise the correct procedure. Sandra had short and tall glasses in the ...
0
votes
0answers
36 views

Find the distance between two points

I'm learning on my own and having some problems understanding these 2 exercises: 1) $$ d = \sqrt{(-\sqrt{6}-\sqrt{3})^2+(0-(-\sqrt{5}))^2} $$ $$ = \sqrt{6+2\sqrt{18}+3+5} = \sqrt{14+2\sqrt{9*2}} ...
-3
votes
3answers
49 views

Find the sum of all products of two distinct naturals, neither exceeding 2015. [on hold]

Find the sum $$(1\cdot2)+(1\cdot3)+(1\cdot4)+\cdots+(1\cdot2015)+(2\cdot3)+(2\cdot4)+\cdots+(2\cdot2015)+\cdots+(2014\cdot2015)$$ any help? I tried with telescope but got nothing
-3
votes
0answers
21 views

solve another system of three equations [on hold]

I have: $x=\dfrac{-.5b-.5c+.25d}{b+c+d}$ $y=\dfrac{.5b\sqrt{3}+.5c\sqrt{3}+.25d\sqrt{3}}{b+c+d}$ $z=b+c+2d$ I need help moving the $b$, $c$, and $d$ to the Left-hand-side; and moving the x, y, and ...
-1
votes
0answers
23 views

solve a system of three equations [on hold]

$x=\dfrac{a-.5c+.25d}{a+c+d}$ $y=\dfrac{.5c\sqrt{3}+.25d\sqrt{3}}{a+c+d}$ $z=a+c+d2$ How do I make it so that only $x$, $y$, and $z$ are on the Right-Hand-Side of the equation while only $a$, $c$, ...
-6
votes
1answer
25 views

coordination questions 123 [on hold]

A ray of light passing through the point $(1, 2)$ reflects on the x-axis at point A and the reflected ray passes through the point $(5, 3)$ find the coordinates of A. Kindly solve full question.
3
votes
1answer
107 views

Find the number $n^{2}$ from the number $\large n^{n^{n^{2}}}$

Find the number $n^{2}$ from the number $\large n^{n^{n^{2}}}$ Any help? I tried with $\log$ but I got nothing.
0
votes
2answers
20 views

Simple algebraic manipulation with 2 equations

My first equations is this: $ d_2 = d - 30.$ My second equations is this: ${1\over d_2 }= {1\over12} - {1\over1.066(d-30)}$ I am trying to solve for $d_2$ in the second equation and then set the ...
-3
votes
0answers
27 views

Determination of polynomial values [on hold]

The polynomial $R(x)=x^4 + Ax^3 + Bx^2 + 10x-1$ ($A,B \in I$) has a remainder of $-15$ when divided $x+1$ and a remainder of $39$ when divided by $x-2$. Determine $A$ and $B$.
1
vote
2answers
68 views

Ways of coloring the $7\times1$ grid (with three colors)

Hints only please! A $7 \times 1$ board is completely covered by $m \times 1$ tiles without overlap; each tile may cover any number of consecutive squares, and each tile lies completely on the ...
0
votes
1answer
48 views

Why is the discriminant of the discriminant negative?

On this link is a question about functions. My question is, in that question itself, a pivotal part of the solution is to realise that the discriminant of the (positive) discriminant is negative. ...
2
votes
1answer
47 views

Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$

Problem : Let $ f: R \to [\frac{1}{2} , 1]$ and $f(x+2) = \frac{1}{2} +\sqrt{f(x) -f(x)^2}$ Then which of the following is always true $(a) f(2) = f(7)$ $(b) f(4) = f(10) $ $(c) f(2) =f(4) $ ...
-1
votes
0answers
15 views

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given? [on hold]

How to derive the relation $T=S_1$ for the equation of a chord of an ellipse whose midpoint is given ? where T = x(x1)/aa + y(y1)/bb - 1 and S1 = (x1)(x1)/aa + (y1)(y1)/bb - 1 where 2a and 2b are the ...
0
votes
2answers
31 views

why dividing a number by 1.25 gives back 20 percent less of original?

So i had to takeout the discount from price. price = 10 discount = 20% my default method has been: price - price*discount ...
3
votes
1answer
55 views

Rewriting $|x-10|+|y-5|\leq 7$ so that absolute values disappear - Algebra

Equation 1: $|x-10|+|y-5|\leq 7$ I want to rewrite this equation into equations that do not have the absolute value. $|A|\leq b$ can be written as $A \leq b$ $A \geq -b$ I want to apply the ...
1
vote
2answers
38 views

Show $n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil$

Let $n$ be a natural number and $b, r > 1$ be two natural numbers, then I guess we have $$ n \cdot \log_b r + \log_b \frac{r}{r-1} \le \lceil n \cdot \log_b r \rceil. $$ where $\lceil x \rceil = ...
4
votes
4answers
98 views

Greatest of the numbers given [duplicate]

To find out the greatest among the number given below: $3^{1/3}, 2^{1/2}, 6^{1/6}, 1, 7^{1/7}$ I have plotted the following graph using graph plotter which is shown below: It can be concluded that ...
-4
votes
0answers
31 views

finding roots of polynomial equation [on hold]

the product of two roots of the equation 4x^2-24x^3+31x^2+6x-8=0 is 1, find all the roots
1
vote
1answer
35 views

Show that $x^2+y^2$ is constant for all values of $\theta$.

Given that $x=3\sin \theta-2 \cos \theta$ and $y=3\cos \theta+2 \sin \theta$ i)Find the value of the acute angle $\theta$ for which $x=y$ ii)Show that $x^2+y^2$ is constant for all values of ...
1
vote
1answer
27 views

Express various trig functions in terms of the sine.

The acute angle $x$ radians is such that $\sin x = k$, where $k$ is a positive constant. Express, in terms of $k$. i) $\sin (2\pi-x)$ ii) $\tan(\frac{1}{2}\pi-x)$ iii) $\cos (\pi+x)$ My attempt: ...
1
vote
2answers
78 views

Is Spivak wrong here, or am I just missing something?

Chapter 1 Problem 18 has the reader doing various proofs with second-degree polynomial functions of the form $x^2 + bx + c$. My issue lies with problem 18d, but it uses knowledge from 18b and 18c, so ...
0
votes
1answer
66 views

algebra question.. [on hold]

If $f : \mathbb{R}\rightarrow \mathbb{R}$, and $f(x)=\frac{2}{4^{x}+2}$ Find the value of $$f\left [ \frac{1}{11} \right ]+f\left [ \frac{2}{11} \right ]+ \cdots +f\left [ \frac{10}{11} \right ]$$
1
vote
1answer
21 views

Sorting triangles by hypotenuse length

I have some points in $xy$ space and I need to sort distances between these points. If I calculate real distance, then I need to perform $\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}$ and this is very time ...
1
vote
6answers
170 views

Why is $\frac{1}{4/3} - \frac{1}{3/2}$ not the same as $\bigl(\frac{4}{3} - \frac{3}{2}\bigr)^{-1}$

If you have the problem:$$\frac{1}{\frac{4}{3}} - \frac{1}{\frac{3}{2}} =?$$ Why can't you change the problem to $(\frac{4}{3} - \frac{3}{2})^{-1}$ and get the same answer? In the first scenario, ...
-1
votes
2answers
27 views

Compound interest 10% per 10 seconds [on hold]

We are starting with 354, ending with 700'000. The interest is 10% every 10 seconds. How long will it take to reach the final figure?
2
votes
3answers
68 views

find the complex number $z^4$

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$. I got that the ...
2
votes
3answers
42 views

Proving $|x+y|=|x|+|y| \iff x\cdot y \geq 0$

Prove: $|x+y|=|x|+|y| \iff x\cdot y \geq 0$. $|x+y|=|x|+|y| \iff x+y=x+y$ and $-(x+y)=-x-y \iff \{x,y\}\geq 0$ and $\{x,y\}\leq 0 \iff x\cdot y\geq 0$ in both cases.
-4
votes
2answers
29 views

Basic root numbers question [on hold]

Hello I want to simplify this expression $1\over\sqrt{(2-\sqrt{5})^2}$ Thank you
1
vote
4answers
136 views

A basic root numbers question

If $\sqrt{x^2+5} - \sqrt{x^2-3} = 2$, then what is $\sqrt{x^2+5} + \sqrt{x^2-3}$?
3
votes
9answers
239 views

How is $x \leq x^2$ false?

There's an equation that says $$x \leq x^2$$ and $x \in \mathbb R$. What I can solve and clearly see is that this equation would be true for any value of '$x$' but then how come my maths teacher ...
0
votes
1answer
58 views

Determining polynomial values

The polynomial has been edited to include the "x" term $R(x)= x^4+Ax^3+Bx^2+10x-1$ has a remainder of $-15$ when divided by $x+1$ and a remainder of $39$ when divided by $x-2$. Determine $A$ and ...
0
votes
1answer
17 views

Quadratic equation roots values was positive but shown as negative

Hi, This screen capture was taken from KhanAcademy. I am an adult learner trying to revisit Algebra I/II concepts. In the video, p was calculated as 1/4 or 4. But, why was is factorized as ...
0
votes
3answers
27 views

Find the number of seven digit whole numbers in which only 2 and 3 are present as digits if no two 2's are consecutive in any number?

Find the number of seven digit whole numbers in which only $2$ and $3$ are present as digits if no two $2$'s are consecutive in any number? My Approach: We can make numbers and see like: ...
0
votes
1answer
31 views

If $500! = 2^m\cdot$N, where N is an odd positive integer, then find $m$

Problem : If $500! = 2^m\cdot$N, where N is an odd positive integer, then find $m$ My approach : Shall we need to expand $500!$ and then find prime factors and see what is the power of 2 in that ...