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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4
votes
1answer
61 views

No. of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$

Number of different real values of $x$ which satisfy $17^x+9^{x^2} = 23^x+3^{x^2}.$ $\bf{My\; Try::}$Using Hit and trial $x=0$ and $x=1$ are solution of above exponential equation. Now we will ...
4
votes
1answer
58 views

Find the inverse function 3

Find the inverse function for the following function: $$f(x) = \log_{\sqrt{4-x^2} } \left(x^3+5x^2-x \right)$$ Thanks.
1
vote
3answers
24 views

Find sum of quartic function coefficients by its plot

the plot of quartic function $y=ax^4-x^2+bx+c$ is given: I need to find a sum of $$\frac{a}{|a|} + 2\frac{b}{|b|} + 4\frac{c}{|c|} $$ How to do it?
0
votes
5answers
38 views

Obtain coefficients of a line from 2 points

I wish to use two points say $(x_1$,$y_1)$ and $(x_2$,$y_2)$ and obtain the coefficients of the line in the following form: $$ Ax + By + C = 0$$ Is there any direct formula to compute.
4
votes
1answer
366 views

What is the Scientific Notation of Zero?

This question was asked here, where the answer uses this description. The last line reads: "The special case of $0$ does not have a unique representation in scientific notation, i.e., ...
4
votes
2answers
65 views

Rationalizing a Denominator with Cube Roots

Rationalize $\dfrac{1}{\sqrt[3]{p^2}+\sqrt[3]{pq}+\sqrt[3]{q^2}}.$ How would I go about doing this without wading through lots of algebra? Is there a trick similar to how you can multiply by ...
0
votes
1answer
28 views

Finding the angle value given 1 point and the centre of a circle

I got the coordinates of the center of a circle $(a,b)$ as well as one other point $(x, y)$. From those I can derive the radius by applying square root to the result of following formula. $$ (x-a)^2 ...
0
votes
3answers
149 views

Best function getting 0 for odd parameter, 1 for even

I'm looking for two functions, assuming x is an integer: $$ f(x) = \begin{cases}0&\text{if x is odd}\\1&\text{if x is even}\end{cases} $$ and $$ g(x) = \begin{cases}1&\text{if x is odd}\\ ...
0
votes
3answers
50 views

Sum of Coefficients in a Polynomial

Find the sum of the coefficients of the terms in the expansion of $(2x+3y-3z)^7$. I know how to do this for binomials, but I was not able to apply the same logic to a trinomial. I believe my other ...
0
votes
2answers
26 views

simple moving average related to a mean

Am I right in this statement? Given a series of numeric values that represent measurements (y) over time (x), the closer a simple moving average is to the mean the less volatility in (y) ?
0
votes
2answers
73 views

simplify expression to find a limit

find the limit of $\large\frac{n}{\sqrt[n]{n!}}$ using the ratio test $$\Large \frac{\left(\frac{(n+1)^{n+1}}{(n+1)!}\right)^\frac{1}{n+1}}{\left(\frac{n^n}{n!}\right)^\frac{1}{n}}$$ I have added ...
1
vote
1answer
28 views

Sum of Coefficients and Number of Terms in Trinomials and Quadrinomials

I already know how to find the sum of coefficients in a binomial, but how do you do it for a trinomial/quadrinomial (after like terms are added)? Example Problem: $(wa+xb+yc+zd)^n$ (all variables are ...
3
votes
4answers
117 views

If we know $x+y+z=1$, $x^2+y^2+z^2=2$, and $x^3+y^3+z^3=3$, how to find $x^4+y^4+z^4$?

Let $x$, $y$, and $z$ be such that $$\begin{align*} x+y+z&=1\\ x^2+y^2+z^2&=2 \\ x^3+y^3+z^3&=3 \end{align*}$$ Then $x^4+y^4+z^4=?$
-2
votes
2answers
84 views

If $\sqrt{n}+ 8= n+1$, what is $n$? [on hold]

If $\sqrt{n}+ 8= n+1$, what is $n$? Please show as many steps as possible so I can understand the process.
0
votes
3answers
78 views

Discriminant of the polynomial $f(x)=4x^3-ax-b$

Definition. The discriminant of the polynomial $f(x)=4(x-x_1)(x-x_2)(x-x_3)$ is the product $16\{(x_2-x_1)(x_3-x_2)(x_3-x_1)\}^2$. How to prove that the discriminant of $f(x)=4x^3-ax-b$ is ...
1
vote
1answer
88 views

Finding the points of intersection of a circle and a line

In a test (of math in arabic language) we we're asked to find the points of intersection of a circle and a line. Their equation is given. In the test I solved system of equations made of their ...
2
votes
1answer
41 views

Is the simplest form of a quadratic equation factored form or standard form?

I've done a bit of research about what defines simplest form, but I could not find a clear answer. Suppose we had to choose: $$(x - 4)(x + 2) \quad\text{or}\quad x^2 - 2x - 8$$ A question asked me, ...
-2
votes
0answers
120 views

Is My Professor Wrong or I Am? [on hold]

In a test (of math in arabic language) we we're asked to find the points of intersection of a circle and a line. Their equation is given. In the test I solved system of equations made of their ...
0
votes
2answers
67 views

How to find all solutions of the equation $\sin x+\cos x=0$ which belong to $(-\pi, \pi)$?

Could you please help me understand and answer this question? Find all  the  solutions of this equation $$ \sin x+\cos x=0 $$ which belong  to  the interval $(-π; π)$ Progress Divided by ...
2
votes
1answer
57 views

Why this approach to differentiate $\log_{10}(x+1)^x$ does not work?

I am trying to differentiate $\log_{10}(x+1)^x$ but I don't get the correct answer, could you please help me? I know that one correct solution is the following: \begin{align} ...
-7
votes
0answers
62 views

hard question, please help [on hold]

11) Assume a sorted array (A) of size n. Propose an algorithm for finding two elements x and y in A that minimize |x-y|. Your algorithm should run in O(n) time for full credit. (Note: |x-y| represents ...
1
vote
3answers
50 views

How to answer the question “what is the domain of this function”?

Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is  the  domain of this function? $$ 2\sin\sqrt{2x-1}+1 $$ ...
-1
votes
2answers
77 views

If we know $a^{1 / 2} + a^{-1/2}$, how can we calculate $a + a^{-1}$?

Someone could help me with this? Thanks so much! Knowing the value of $a^{1/2}+a^{-1/2}$, calculate $a+a^{-1}$.
5
votes
1answer
60 views

What are some remarkable and interesting uses of AM-GM Inequality ? Cite and explain with problems.

There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications. What I want is a collection of questions/problems which look very complex but get ...
1
vote
1answer
61 views

2014 Fall OMO #28

Here is a problem from this year’s OMO: Let $S$ be the set of all pairs $(a,b)$ of real numbers satisfying $1+a+a^2+a^3 = b^2(1+3a)$ and $1+2a+3a^2 = b^2 - \frac{5}{b}$. Find $A+B+C$, where $$ A = ...
3
votes
1answer
25 views

inequality on real numbers.

Suppose one is given real numbers $\alpha_1, \lambda_0 \ge1$ and $ \lambda_1$ such that $\alpha_1^2\le1$ and $\lambda_0^2=1+\lambda_1^2$. Then it is easy to show that ...
-7
votes
0answers
38 views

need a pre calc answer ASAP on tangent functions [on hold]

My pre calculus teacher is making us use her way of tangent function graphing using Undefined, (-1, 0, 1, Undefined so basically finding your maximas and minimas which is easy enough for sin/cos ...
0
votes
1answer
30 views

$\mathbb E[\bar X_n]=0$

A conditional normal rv sequence, does the mean converges in probability, in this question how can i get $\mathbb E[\bar X_n]=0$? Here is my attempt; $$\mathbb E[\bar ...
8
votes
3answers
136 views

How to solve the differential equation $(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$?

Solve $$(2x^3y)\:\text{dy}+(1-y^2)(x^2y^2+y^2-1)\:\text{dx}=0$$ I tried the substitution $y^2=t$ ; $2y\:\text{dy}=\text{dt}$ to get $$(x^3)\:\text{dt}+(1-t)[(x^2+1)t-1]\:\text{dx}=0$$ ...
1
vote
2answers
50 views

Closed form for $\sum_{t=0}^{n} t^2x^t$

I am trying to come up with a formula for $\Sigma_{t=0}^{n} t^2x^t$ I understand that $$\sum_{t=0}^{n} x^t=\frac{1-x^{n+1}}{1-x}$$ I also was able to find that $$\sum_{t=1}^{n} ...
-3
votes
0answers
19 views

Finding a quadratic function to determine how fast someone travelled [on hold]

I need to determine a quadratic function that would determine how fast a person was cycling given that they ran $6$ miles and then bicycle $20$ miles. If the bicycle speed is $8$ miles per hour faster ...
4
votes
1answer
55 views

Inequality $\frac{x^3+y^3}{x-y}>4$

Let $x>y>0$ and $xy\geq 1$. Prove that $$\frac{x^3+y^3}{x-y}>4.$$ Of course we can factor $(x^3+y^3)=(x+y)(x^2-xy+y^2)$, but it is not very useful. For fixed $x-y$, we can try to find the ...
1
vote
2answers
95 views

How does $n < 2^n$ become $\log n < n$ by taking log of both sides?

How does $n < 2^n$ become $\log n < n$ by taking the log of both sides? I understand the left side but I do not understand the right side of the inequality. The once was $\log 2^n$ becomes $n$ ...
3
votes
1answer
29 views

Project Motorola: setting up and solving an equation

Stuck on a homework project in a highschool college algebra question. I'm given the following information: Tact time is the average time to pick and place one part. Throughput is the number of ...
0
votes
2answers
29 views

Can this be rewritten as the following? [on hold]

Can $x(x^2-1)-1(x-1)$ be rewritten as $(x-1)(x^2-1)(x-1)$ ? It is during decomposition in factoring. Thanks.
1
vote
1answer
44 views

Partial Fraction Decomposition of Exponential Generating Functions

I want to see if it is possible to write $$ \left(\frac{x}{e^x-1}\right) \left(\frac{x^2/2! }{e^x-1-x}\right) \left(\frac{x^3/3!}{e^x-1-x-x^2/2}\right)$$ as a linear combination of the factors ...
3
votes
2answers
89 views

How did Sir Isaac Newton develop and formulate the famous binomial theorem?

After completing combination, I have started to read Binomial Theorem. My book only mentioned about Pascal's Triangle. And the formula was then given straightforward. But how did Sir Issac Newton ...
2
votes
7answers
572 views

How to solve these equations for x and y..

equations are $(x-y)(x+2y)(2x+y) = 20$ and $x^2+xy+y^2 = 7$ i want the METHOD not the solutions
0
votes
1answer
31 views

Defining a list using set theory?

EDIT: Changed set If I have the following set of numbers: $\{1, 2, 4, 8, 16,...\}$ where the universe of discourse is natural numbers $\{0, 1, 2, ...\}$ How can I define this? I note that ...
-7
votes
0answers
49 views

how to factor this expressiion [on hold]

How to factor $\frac{2x^2-4x}{x+10}$ ?
8
votes
1answer
58 views

$(1-a)(1-b)(1-c)(1-d)\geq abcd$ for $a^2+b^2+c^2+d^2=1$

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Prove that $$(1-a)(1-b)(1-c)(1-d)\geq abcd.$$ I thought about substituting $a=\sqrt{w},b=\sqrt{x}$, etc. (assuming first that $a,b,c,d$ ...
1
vote
1answer
28 views

Solution for a complexed equation

Find $z$ for the equation $e^z + e^{-z} = 0$. So $$e^z + e^{-z} = 0 \iff e^z = -e^{-z} \iff e^z = e^{\pi i - z} \iff z = \pi i -z + 2\pi ik$$ I understand all expect the $2\pi ik$. Can you ...
1
vote
1answer
26 views

find roots in the complexes

Find the roots of: $$ z^2 -3z +4iz = 1-5i $$ Rearranging the terms: $z^2 + z(4i-3) + 5i - 1 $ Solving by using the quadratic formula: $$z_{1,2} = \frac{3-4i\pm \sqrt{(4i-3)^2 -4(5i-1)}}{2}$$ ...
9
votes
1answer
113 views

Prove that $ ax^2+bx+c=0 $ has at least one root in $(0,1)$ if $10a+12b+15c=0$

If $10a+12b+15c=0$, Prove that $$ ax^2+bx+c=0 $$ has at least one root in $(0,1)$. Progress I tried to solve this by Rolle`s theorem ($f'$ has a root between any two roots of $f$), but could not ...
3
votes
1answer
31 views

Inequality $a^2b^2+2(a+b)\geq 4ab+1$

Let $a,b\geq 1/2$. Prove that $$a^2b^2+2(a+b)\geq 4ab+1.$$ We know that $(ab-1)^2\geq 0$ implies $a^2b^2+1\geq 2ab$, so the inequality reduces to $2(a+b)\geq 2ab+2$, or $a+b\geq ab+1$. But this is ...
4
votes
2answers
20 views

Evaluating $\sum_{i=a+1}^{N}\frac{i(i-1)}{i-a}$

I am trying to solve the German Tank Problem. There might be numerous ways of finding the expected value of N. However, the way in which I am proceeding, I need to find this sum. However I am stuck at ...
6
votes
1answer
64 views

Given $f(x)$ and $g(x)$, find $(fg)(x)$

I've attempted to solve the problem below, and here is what I got for a solution: Given $f(x)=x^2-9$ and $g(x)=x^2+3x-1$, find $(fg)(x).$ $$ \begin{align} (fg)(x)&=(x^2-9)(x^2+3x-1)\\ ...
5
votes
6answers
105 views

Solve $\sin2x +\sin x = 0$ algebraically

I am studying for a final and came across a review question that I have no idea how to do. The question is "Solve the equation $\sin(2x) + \sin(x) = 0$ on the interval $[0, 2\pi)$. I can graph it ...
1
vote
2answers
32 views

Is parametric form of a given function unique? [on hold]

Can we say that for any given function in single/multivariable, it is always possible to have a parametric form? (Elementary functions, complicated functions?) Given any function, is parametric form ...
1
vote
0answers
14 views

Macaulay duration for a coupon bond. Proof

I am working on showing the following. There is a coupon bond redeemable at par with annual coupon rate $r$ per year. The yield to maturity is $i$. The total number of coupons is $n$. Show ...