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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166 views

Compound interest for retirement

Determine how much you will have to save each month at $3$, $6$, $9$, and $12$ percent compounded monthly for you to accumulate a nest egg for retirement. The variables are current age, age of ...
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1answer
94 views

What will be the minimum value of $\frac{p^2}{\tan9^\circ} + \frac{q^2}{\tan27^\circ} + \frac{r^2}{\tan63^\circ} + \frac{s^2}{\tan81^\circ}$?

What will be the minimum value of $$\frac{p^2}{\tan9^\circ} + \frac{q^2}{\tan27^\circ} + \frac{r^2}{\tan63^\circ} + \frac{s^2}{\tan81^\circ}$$ if $$p+q+r+s=5$$ where $p, q, r, s$ are positive reals? ...
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4answers
114 views

Easy way to find roots of the form $qi$ of a polynomial

Let $p$ be a polynomial over $\mathbb{Z}$, we know that there is an easy way to check if $p$ have rational roots (using the rational root theorem). Is there an easy way to check if $p$ have any roots ...
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1answer
148 views

Division is equal to zero

I have this: $f(x) = 0$ where $f(x) := \cfrac{3x^2 - 5x + 2}{x + 2}$ How do I solve that? Do I multiply by $(x + 2)$ and solve $3x^2 - 5x + 2=0$ or solve $3x^3 + x^2 - 8x + 4=0$ with Horner ...
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173 views

Logical question problem

A boy is half as old as the girl will be when the boy’s age is twice the sum of their ages when the boy was the girl’s age. How many times older than the girl is the boy at their present age? This ...
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0answers
63 views

What can be the least possible pair of naturals $(a, b)$ in the following case?

$a, b$ are two naturals such that, $$a^{2}-b^{2}=k^{3}$$ and $$a^{3}-b^{3}=c^{2}$$ where $k^{3}$ and $c^{2}$ are perfect cube and square respectively. What can be the least possible pair of ...
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2answers
351 views

$\sqrt{x} +y = 4$, $x+ \sqrt{y}= 6$, find the solution $(x,y)$

$\sqrt{x} +y = 4$, $\sqrt{y} +x= 6$, find the solution (x,y). $NOTE$ : $\sqrt{4}+1= 4-1$, $\sqrt{1} +4 =1+4$
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1answer
421 views

How to resize an image?

I am not sure about the title of this question, so if someone knows an appropriate one, please rename it. It's a programming related question (but doesn't involve any programming). I posted it on ...
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1answer
61 views

Working out interest rates on a futures contract/exchange rates contract.

okay my main problem is I have to work out a $3$ month interest rate for the us dollar. The question I'm stuck on is At the end of trading on $1$ January $2012$ the dollar/pound spot exchange rate ...
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2answers
131 views

How do I proceed with these quadratic equations?

The question is $$ax^2 + bx + c=0 $$ and $$cx^2+bx+a=0$$ have a common root, if $b≠ a+c$, then what is $$a^3+b^3+c^3$$
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568 views

Definition of quadratic equation?

What is a quadratic equation and what is its simplified and cannonic form?
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1answer
64 views

What is the number of all possible values of $[Z^{6}]$?

Its given that $$[Z]=3$$ $$[Z^{2}]=11$$ $$[Z^{3}]=41$$ Then, what is the number of all possible values of $[Z^{6}]$ where $[\;\cdot\;]$ is floor function.
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1answer
125 views

Prove that $\frac{x^5-x^2}{x^5+y^2+z^2}+\frac{y^5-y^2}{x^2+y^5+z^2}+\frac{z^5-z^2}{x^2+y^2+z^5}≥0 $.

Given $x, y, z $ are 3 positive reals such that $xyz≥1$. Prove that $$\frac{x^5-x^2}{x^5+y^2+z^2}+\frac{y^5-y^2}{x^2+y^5+z^2}+\frac{z^5-z^2}{x^2+y^2+z^5}≥0.$$ This question is so complicated. I failed ...
2
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1answer
121 views

Simplifying $|a+b|^2 + |a-b|^2$

I want to simplify $|a+b|^2 + |a-b|^2$ where $a, b \in \mathbb{C}$. I've used Wolfram Alpha to get $$ |a+b|^2 + |a-b|^2 = 2\left(|a|^2 + |b|^2\right) $$ I'm trying to understand the steps involved in ...
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3answers
865 views

Write each pair of equations as a single equation in $x$ and $y$.

Write each pair of equations as a single equation in $x$ and $y$. a)$\begin{cases} x=t+1 &\\ y=t^2-t & \\ \end{cases}$ b)$ \begin{cases} x=\sqrt[3]{t}-1 &\\ ...
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3answers
488 views

Changing from quadratic formula to standard form.

The graph of a quadratic function has $x$-intercepts $-1$ and $3$ and a range consisting of all numbers less than or equal to $4$. Determine an expression for the function. This is my problem. I ...
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3answers
6k views

Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$.

Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ where a) $f(x)=2x+3,$ b) $f(x)=\frac{1}{x+1},$ c) $f(x)=x^2.$ I believe that if anyone can help me out with the first one, the ...
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2answers
185 views

Pythagorean Triplets with “Bounds”

I am interested in the algebraic/geometric way of finding the pythagorean triplets such that $$a^2 + b^2 = c^2$$ $$a + b + c = 1000$$ I do the obvious $$a + b = 1000 - (a^2 + b^2)^{1/2}$$ $$a^2 + ...
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4answers
891 views

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$.

Solve $\cos^{n}x-\sin^{n}x=1$ with $n\in \mathbb{N}$ I have no idea how to deal with this crazy question. One idea came into my mine is factorization, but I can't go on... Can anyone help me please? ...
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4answers
2k views

Find the domain of $f(x)=\frac{3x+1}{\sqrt{x^2+x-2}}$

Find the domain of $f(x)=\dfrac{3x+1}{\sqrt{x^2+x-2}}$ This is my work so far: $$\dfrac{3x+1}{\sqrt{x^2+x-2}}\cdot \sqrt{\dfrac{x^2+x-2}{x^2+x-2}}$$ $$\dfrac{(3x+1)(\sqrt{x^2+x-2})}{x^2+x-2}$$ ...
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2k views

A Tri-Factorable Positive integer

Found this problem in my SAT book the other day and wanted to see if anyone could help me out. A positive integer is said to be "tri-factorable" if it is the product of three consecutive integers. ...
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3answers
191 views

Really stuck on strategy for this fundamental theorem of algebra type problem.

I am at a loss at how to approach this problem. It doesn't make sense to me to solve without finding the roots. I couldn't find the roots without a calculator anyway, but you're not meant to. Show ...
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4answers
70 views

graphing and functions

one of my problem says that the average slope formula is: $\dfrac{f(x_2)-f(x_1)}{x_2-x_1}$ It tells us the average slope over the interval from $x_1$ to $x_2$ Then it says to take the average slope of ...
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3answers
869 views

why 0=0 is not possible??

Hi one of my friend showed me one proof i.e. $1)$ $2^2 - 2^2 = 10 - 10$ $2)$ $(2+2) (2-2) = 5 (2-2)$ $3)$ dividing both sides by (2-2) $4)$ $(2 + 2) = 5$ I know this is wrong in first line as ...
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2answers
121 views

Product of Polynomials

If $p$ is a polynomial of degree $n$ and $q$ is a polynomial of degree $m$, then their product $p \cdot q$ is given by: $$ (p \cdot q)(x) = \sum_{i = 0}^{n + m} \left ( \sum_{k = 0}^i p_k q_{i - k} ...
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1answer
89 views

Simultaneous equations, two unknowns

I really should've paid more attention to maths in school... I have some fairly simple simultaneous equations in the following format. VMax = DMax + (DMax - DMin) * GMax VMin = DMin - (DMax - ...
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3answers
226 views

How to show x and y are equal?

I'm working on a proof to show that f: $\mathbb{R} \to \mathbb{R}$ for an $f$ defined as $f(x) = x^3 - 6x^2 + 12x - 7$ is injective. Here is the general outline of the proof as I have it right now: ...
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3answers
156 views

Proving $\sqrt{n}+\frac{1}{\sqrt{n+1}} \geq \sqrt{n+1}$

I would like to know how to prove the following assertion : For every $n>0$: $$\sqrt{n}+\frac{1}{\sqrt{n+1}} \geq \sqrt{n+1}$$
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1answer
215 views

KhanAcademy-Trig Identities 1

I'm going through Khan Academy and I'm stuck at Trig Identity and there is something I don't understand. Given that $x$ is in the first quadrant and $\sec x$ is $\frac{2\sqrt3}3$, what is $\cos ...
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2answers
80 views

Am I right in thinking $\frac{x^{2}}{ax+b}$ is an improper rational expression?

Am I right in thinking $\dfrac{x^{2}}{ax+b}$ is an improper rational expression? If so, can someone help me figure out how to write it as the sum of a polynomial and proper rational expression? I ...
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3answers
998 views

Pre Calculus and Rate of Change

Sacha drains the water from a hot tub. The hot tub holds $1600$ L of water. It takes $2$ h for the water to drain completely. The volume of water in the hot tub is modelled by $V(t) = 1600 - ...
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892 views

is division by zero automatically irrational?

I know that division by zero is undefined and, also not rational, but I am not sure that this means its default status becomes irrational because of this. Can anyone clarify?
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2answers
479 views

What are the benefits of using $x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$ to solve quadratic equations?

When I was in high school, they taught me to solve quadratic equations with this formula: $$x=\frac{\sqrt{4 \text{ac}+b^2}-b}{2 a}$$ EDIT: The original formula is this one: $x = \dfrac{-b \pm ...
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2answers
61 views

Distributing an item equally.

For the following question A $10$ foot plank of wood is cut to give three equal lengths with a shorter length left over. Which is more a)The length of one of equal pieces ...
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2answers
167 views

Sum and difference of sines and cosines

How would I solve the following question? Show that $$2\sin(127.5)\sin(97.5)=(\sqrt{3}+\sqrt{2})/2$$ My work is I know $$\sin A\sin B=(-1/2)(\cos(A+B)-\cos(A-B))$$ So I did ...
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1answer
67 views

Solving for unknown, trouble with $\ln$ and $\exp$

Having some trouble understanding $\ln$ and $\exp$ rules and what to do in this situation. Perhaps it has just been a very long day... $$\hat{Y} = \exp \left[\left(\hat{\beta_0} + \sum_i ...
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3answers
1k views

Calculating the average speed using only velocities

For the following question: Pedro travels by bus to school at an average speed of $40$ km/hr. He is driven home by the same route by a friend's car at an average speed of $50$ km/hr. Which of ...
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1answer
44 views

Constructing an equation using proportionality

The question is: If the rate of a certain chemical reaction doubles for every $10$ degree rise in temperature then which is greater: a) Twice the rate at $10$ degrees b) Half the rate ...
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1answer
1k views

Find distance traveled by tips of hands of clocks?

The short and the long hands of a wall clock are $8$ cm and $12$ cm respectively. Find the sum of the distance traveled by their tips in $3$ days. Give your answer in terms of $\pi$. My ...
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1answer
120 views

Number Line Question

I am stomped on the following question Which is greater judging from the number line if $\rm JL = KM.$ a) $\rm JK$ b) $\rm LM$ (Answer : Both are the same) I would like to know ...
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3answers
67 views

How does this expand out?

I'm finding myself getting back into math related stuff for the first time in a while. So please be patient with me. How does $\frac{(n-i)(n-i+1)}{2}$ expand out to: $\frac{n^2 - (2i - 1)n - i + ...
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3answers
1k views

Quadratic equation with absolute value

Prepping for the GMAT, I came across the following question: What is the product of all solutions of: $$x^2 - 4x + 6 = 3 - |x - 1|?$$ First, I set up two equations, ie: $$x^2 - 4x + 6 ...
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1answer
143 views

I need help simplifying and reorganizing this algebraic equation

I've developed the following algebraic equation which could probably be simplified further. Also, I need it reorganized to solve for x and Y (in terms of A, x, and Y. Not looking for a numerical ...
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5answers
600 views

How to disprove there exists a real number $x$ with $x^2 < x < x^3$

I realize that the only method is to show various cases: I must test for $x > 1$, $x < -1$, $0 \leq x \leq 1$, and $-1\leq x \leq0$. But even with this, I don't understand how to inject the ...
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2answers
98 views

What's wrong with this conversion?

I need to calculate the following limes: $$ \lim_{n\rightarrow\infty} \sqrt{\frac{1}{n^2}+x^2} $$ My first intuition was that the answer is $x$, but after a bit of fiddling with the root I got ...
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11answers
3k views

Solving $5^n > 4,000,000$ without a calculator

If $n$ is an integer and $5^n > 4,000,000.$ What is the least possible value of $n$? (answer: $10$) How could I find the value of $n$ without using a calculator ?
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4answers
2k views

Sum of n different positive integers is less than 100. What is the greatest possible value for n?

I am stumped on the following question: The sum of n different positive integers is less than 100. What is the greatest possible value for n? a) 10, b) 11, c) 12, d) 13, e) 14 ...
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3answers
154 views

What is a good technique for solving polynomials?

Say for example: $6x^{3}-17x^{2}-4x+3=0$ I sort of look at it and don't know where to start, other than just guessing what the first one would be and trying to do from there. Is there a good ...
3
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5answers
268 views

Proving Quadratic Formula

purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become ...
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1answer
356 views

Word-Problem using a bar-chart

I can not manage to solve this problem. Any suggestions on how I can solve it. If in 2006 Pharmacom spent the same dollar amount on administration (administration outgoings) as in 2005, but the ...