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
4answers
598 views

$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ show that $x=-c/b$ when $a=0$

OK, this one has me stumped. Given that the solution for $ax^2+bx+c =0$ $$x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}\qquad(*)$$ How would you show using $(*)$ that $x=-c/b$ when $a=0$ (Please dont use $a=0$ ...
5
votes
2answers
511 views

Number of positive integral solutions for $ab + cd = a + b + c + d $ with $1 \le a \le b \le c \le d$

How many positive integral solutions exist for: $ab + cd = a + b + c + d $,where $1 \le a \le b \le c \le d$ ? I need some ideas for how to approach this problem.
2
votes
2answers
165 views

Reducing fractions?

I want to reduce the two following fractions: $$ \frac{2x + 2y}{x + y} $$ $$ \frac{3ab^2}{12ab} $$ I fully understand the concept of reduce fractions of this type: $$ \frac{15}{20} $$ but i ...
3
votes
1answer
366 views

Is $(x+1)^2 = (x+1)^3$ for any $x$?

This is something I've been struggling to understand since the past few days. Let's take an example: Prove/Disprove: $(x+1)^2 = (x+1)^3$ for all real values of $x$. Proof: Let us assume the ...
11
votes
4answers
2k views

What is the term for a factorial type operation, but with summation instead of products?

(Pardon if this seems a bit beginner, this is my first post in math - trying to improve my knowledge while tackling Project Euler problems) I'm aware of Sigma notation, but is there a function/name ...
6
votes
3answers
113 views

Numbering students inequality problem

Ten students are sitting around a campfire. A teacher randomly assigns each student a different number from 1-10. Another teacher assigns a new number to each student with the requirement that the new ...
5
votes
2answers
155 views

Solving an Exponential Equation

$$3^{x-1}+3^{x-2}+3^{x-3}=3159$$ Another exponential equation I'm having a hard time with, the answer is given and equals to : $8$. I'm absolutely sure I'm making a wrong step somewhere along the ...
2
votes
4answers
101 views

How to “efficiently” compute the number of solution of $25x= 5y+8z$ such that $x,y,z \in [0,9]$ and $x,y,z \in \mathbb{W}$?

The mother problem: Find the sum of all $3$ digit numbers which are equal to $25$ times the sum of their digits. So we can write: $$\begin{align} 100x+10y+z &= 25 ...
1
vote
3answers
464 views

Solving an exponential inequality

$$(0{,}25)^{3-0{,}5x^2}\leq8$$ Answers given are: $[-3;3]$ Below is where I got with this, I'm pretty sure I took a wrong approach here. Any help at all is appreciated. $$\begin{aligned} ...
2
votes
2answers
3k views

How to set up these ratios?

Here's a question while reading my textbook: For about 10 years after the French Revolution, the French government attempted to base measures of time on multiples of ten: One week consisted of ...
5
votes
1answer
202 views

How many triplets of real numbers $(x, y, z)$ which satisfy these $3$ restriction:

How many triplets of real numbers $(x, y, z)$ which satisfy : $$(x + y)^3 = z$$ $$(y + z)^3 = x$$ $$(z + x)^3 = y$$ I need some approaches for solving this problem.
1
vote
1answer
77 views

Solving an Exponential Equation

I'm having a hard time with this exponential equation, I'm sure I'm doing some kind of a "minor" mistake again somewhere along the way. Your help is very much appreciated. $$4\cdot4^{2x}-9\cdot ...
2
votes
4answers
180 views

Solution for a quadratic equation

The following is the solution from my book, but without the calculation path. $$ \frac{(18 \pm \sqrt{-180})}{6} = 3 \pm\sqrt{-5} $$ Why is it $-5$?
2
votes
1answer
118 views

Solving an Exponential Equation

Answer is given, and it equals to 1. $$ 2\cdot 16^{x}-2^{4x}-4^{2x-2}=15 $$ $2^{4x-4}=-6,5$ <- This is where I reached, which is clearly wrong
3
votes
4answers
735 views

How many distinct real roots does $ (x^2 + x – 2)^3 + (8–2x^2 )^3 = (x^2 + 3x + 2)^3 $have?

How many distinct real roots does $ (x^2 + x – 2)^3 + (8–2x^2 )^3 = (x^2 + 3x + 2)^3 $have? If I didn't make any mistake that equation could be reduced to the form:$$(2+6x^2)(x+2)^3=(8–2x^2 ...
2
votes
2answers
3k views

Find 5th degree polynomial equation from points?

I have a set of estimated points, listed below, and I would like to know how to calculate the polynomial. I've looked at Wikipedia, and I don't quite understand how at works. Example 1 on ...
-2
votes
1answer
456 views

Upper bound to lower bound

Let $x(a)\geq1$ and $y(b)\geq1$. I have a relation $x(a) \leq k(a,b)y(b)$ for all $k(a\geq b) \geq 1$ and $x(a)=y(b)$ when $k(a=b)=1$. Can we conclude that $x(a)\geq y(b)$ ?
1
vote
1answer
840 views

Hard simultaneous equation problem $5x^2y-4xy^2+3y^3-2(x+y)=0$, $xy(x^2+y^2)+2=(x+y)^2$

How to solve this system of equations: $$\begin{align*} 5x^2y-4xy^2+3y^3-2(x+y) &=0 \\ xy(x^2+y^2)+2 &=(x+y)^2 \end{align*} $$
1
vote
2answers
313 views

How to find the minimum value of $\frac{x}{2x+3y}+\frac{y}{y+z}+\frac{z}{z+x}$?

Let $x,y,z\in [1,4]$ such that $x \geq y$ and $x \geq z$. Find the minimum value of this expression: $$ P=\frac{x}{2x+3y}+\frac{y}{y+z}+\frac{z}{z+x} $$
0
votes
2answers
154 views

Equation manipulation question

I have an equation: $$2^{x - 1} = \frac{360}{y}$$ I want to manipulate it so that $x$ is on the LHS of the equal sign, all by itself. Do you think I remember how to do that? Any ideas?
0
votes
1answer
127 views

Geometric progression - How many members? a=7, r=4, Sum=7340025

In Geometric Progression, first member is equal to 7 and the common ratio is equal to 4. How many members should be taken in to the sequence that it would amount to 7340025?
1
vote
4answers
218 views

How many ways to form a committee, subject to certain restrictions?

From among six couples, a committee of $5$ members is to be formed. If the selected committee has no couple, then in how many ways can the committee be formed? My approach is to count things ...
6
votes
5answers
1k views

Algebraic proof that collection of all subsets of a set (power set) of $N$ elements has $2^N$ elements [duplicate]

In other words, is there an algebraic proof showing that $\sum_{k=0}^{N} {N\choose k} = 2^N$? I've been trying to do it some some time now, but I can't seem to figure it out.
2
votes
1answer
4k views

symmetry with respect to the x-axis and the y-axis

Please help me prove that if a graph is symmetric with respect to the x-axis and to the y-axis, then it is symmetric with respect to the origin.
3
votes
2answers
205 views

Inserting numbers to create a geometric progression

Place three numbers in between 15, 31, 104 in such way, that they would be successive members of geometric progression. PS! I am studying for a test and need help. I would never ask anyone at ...
1
vote
2answers
110 views

Struggling with a question at high school level

Jack has 40 dollars more than Jill. Jack gives 20 dollars to Jill. Then they have the same amount of money. I found the answer by method of exhaustion, but how can this be solved more elegantly?
3
votes
1answer
177 views

Arithmetic progression

I'm studying for a test, and I am having a hard time with this particular exercise. The first member is equal to 7 and the fifth member is equal to 59. How many members should be taken in to the ...
2
votes
2answers
498 views

Math of Human Pyramids

I was writing a blog post today, and I ended up asking the question of how many layers tall a human pyramid would be if it contained all of the people who use Facebook, approximately 750 million. ...
4
votes
8answers
2k views

Maximizing the sum of two numbers, the sum of whose squares is constant

How could we prove that if the sum of the squares of two numbers is a constant, then the sum of the numbers would have its maximum value when the numbers are equal? This result is also true for ...
3
votes
6answers
325 views

Optimizing $a+b+c$ subject to $a^2 + b^2 + c^2 = 27$

If $a,b,c \gt 0$ and $a^2+b^2+c^2=27$, find the maximum and minimum values of $a+b+c$. How to solve this one? (Here's the source of inspiration for the problem.)
-3
votes
1answer
138 views

Creating Formula for a function $Y$ = $f(X)$ satisfying some conditions [closed]

Anyone here who can give me some help about creating formula with these rules? $X$ is given to find $Y$ $Y ≤ X$ $Y$ cannot be equal $0$ -
4
votes
1answer
58 views

Solve for variable

$$0 = \frac2{r-1}-\frac3{r+4}+\frac1{r+5}$$ So to my understanding I could give them all the same denominator by multiplying their denominators with each others denominators and numerators. Or could ...
0
votes
1answer
79 views

How to approach this problem?

The problem statement: The expenses of a tuition class are partly fixed and partly variable with the number of students.The charge is $40\$$ per head when there are $25$ students and $60\$$ ...
2
votes
2answers
95 views

Difference of Logarithms to form a quotient?

Write as a single logarithm: $\log_8(5) - 2\log_8(6)$ To my understanding; because they are the same base you can just evaluate $\log_8\left(\frac{\log(5)}{\log(6)}\right)$ which is shown on the ...
6
votes
5answers
542 views

Prove this number fact

Prove that $x \neq 0,y \neq 0 \Rightarrow xy \neq 0$. Suppose $xy = 0$. Then $\frac{xy}{xy} = 1$. Can we say that $\frac{xy}{xy} = 0$ and hence $1 = 0$ which is a contradiction? I thought ...
2
votes
1answer
164 views

Can you solve for a value within a floor function?

I'm writing some software which performs activities using an exponential back-off delay e.g. performs an action at t = 1, 2, 4, 8, 16 etc, assuming a base of 2. I want the base to adjust dynamically ...
1
vote
3answers
200 views

Continued fraction form for rational numbers less than $1$

How could we convert a rational number (less than $1$) to the continued fraction form? This is probably an extension of this question. After reading Bill Dubuque's answer here and here, I got ...
2
votes
3answers
256 views

Algebra in trigonometry, algebraic proof?

The picture says it all. "Vis at" means "show that". My first thought was that h is 2x, which is not correct. Maybe the formulas for area size is useful? EDIT: (To make the question less dependent ...
4
votes
4answers
155 views

A “fast” and “manual” approach for sorting a set of fractions

What could be the fastest manual approach for sorting (ascending) these fractions: $$\frac{117}{229},\frac{143}{291},\frac{123}{241},\frac{109}{219}$$ I would also be very grateful if somebody ...
6
votes
3answers
390 views

A “fast” way to find the sum of the sequence $5,5.5,5.55,5.555,5.5555,\ldots $ (20 terms)

My initial approach is diving the whole sum by $9$ and taking the common $5$ out which gives $$\frac{5}{9}[(10-1)+(10-0.1)+(10-0.01)+\cdots + (10-10^{-19})]$$ after some algebra this could be reduced ...
1
vote
2answers
87 views

Question about a Question: Simplifying Fractions

In a question I asked several weeks ago an interim step reached was a.): $$\frac{1}{(x-6)!6!}=\frac{1}{(x-4)!4!}$$ hence b.): $$ \frac{(x-4)!}{(x-6)!}=\frac{6!}{4!}$$ I'm not following how we got ...
6
votes
2answers
900 views

Challenging inequality: $abcde=1$, show that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}+\frac{33}{2(a+b+c+d+e)}\ge{\frac{{83}}{10}}$

Let $a,b,c,d,e$ be positive real numbers which satisfy $abcde=1$. How can one prove that: $$\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} +\frac{1}{e}+ \frac{33}{2(a + b + c + d+e)} ...
0
votes
3answers
314 views

When does this quadratic equation have a real solution?

While solving some quantitative problem,I got stuck in this one: For what values of $m$ is $y = 0$, if $y=x^2+(2m+ 1)x+m^2-1$? ($x$ is a real number). Could anybody help me to understand this ...
0
votes
1answer
53 views

How do I rewrite the expression $r+(3/2)s-(1/3)t$

Can someone please explain how to rewrite the expression $r+(3/2)s-(1/3)t$ in terms of $a, b,$ and $c$, where $r= \log (a), s=\log(b) \text{ and } t=2\log(c)$. The rewritten expression should be in ...
1
vote
3answers
516 views

Equation for the family of lines that passes through $3y-5x-10=0$ and $3y-\frac{x}{3}-\frac{5}{3}=0$

Find an equation for the family of lines that passes through the intersection of $3y-5x-10=0$ and $3y-\frac{x}{3}-\frac{5}{3}=0$
1
vote
1answer
248 views

How to solve these inequalities?

How to solve these inequalities? If $a,b,c,d \gt 1$, prove that $8(abcd + 1) \gt (a+1)(b+1)(c+1)(d+1)$. Prove that $ \cfrac{(a+b)xy}{ay+bx} \lt \cfrac{ax+by}{a+b}$ Find the greatest value ...
1
vote
1answer
92 views

Calculating a relative offset, given two different endpoint measurements and a midpoint target value

I have a packetized binary data file that can be referenced in one of two ways: By an arbitrary time tick counter, and By a file offset in bytes. In my example file, I have the following packet ...
8
votes
4answers
1k views

Easy ways to find partial fraction representation? (via a concrete example)

In a homework assignment (about generating functions) the students find themselves having to expand $\frac{3-7x+9x^{2}-3x^{3}}{\left(1-x\right)^{4}}$ intro partial fractions. Using some automated tool ...
1
vote
1answer
66 views

Curious about how to work out a formula to solve a simple scenario

I have 838 rows of data which each have a calculated column showing me the duration from creation to resolution. The average of all those rows duration is 63 hours. I want to say if I move x rows up ...
0
votes
1answer
357 views

How to normalize multiple vectors?

I have an Excel spreadsheet with multiple vectors of 3 numbers. I would like to normalize those vectors so that they are within the same range. ...