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
2answers
164 views

solving a problem using degrees OR radians

hey so i'm programming something that finds an angle of a line between 0 and 180 degrees based on two points.... the equation to find the answer is ...
4
votes
3answers
348 views

Fractions with radicals in the denominator

I'm working my way through the videos on the Khan Academy, and have a hit a road block. I can't understand why the following is true: $$\frac{6}{\quad\frac{6\sqrt{85}}{85}\quad} = \sqrt{85}$$
2
votes
2answers
263 views

question about clocks

suppose we have two clock A and B. each hour A clock outruns by 2 minute then B which always shows correct time.in 1 January both clock was corrected and it was showed 16:00 ...
1
vote
2answers
464 views

Formula for inverse relationship between 'scale' and 'physical size'

I'm having trouble solving the following problem. Hopefully someone can help! I have some drawing plans to prepare: the orginal drawing is a scale 1:200 (printed at 100%) if i print at 50% it will ...
1
vote
3answers
168 views

How do I calculate where to plot a point on a document, given other reference points?

Imagine a PDF document. This document has been geo-referenced - I have been given two (x,y) coordinates that correspond to two (lng/lat) pairs. Based on this data, I need to plot a different pair of ...
2
votes
6answers
270 views

Factoring Quadratics

Is there a method to find which numbers to use when simplifying quadratics? For example $x^2 + 5x + 6$ is easy enough to find, but what if I have bigger numbers, or I have this quadratic expression: ...
0
votes
2answers
2k views

How to solve this problem using set theory?

$36$ students took English and Math test. $25$ passed English, and $28$ passed Math. $20$ passed both subjects. a. How many students failed both subject? b. How many students passed english only? c. ...
2
votes
2answers
82 views

Need help solving this basic algebra question (addition and division)

Believe it or not I graduated with a BSc in Computing Science, but apparently that means nothing after being out of school for a year. The question is: $(c+n)/(t+n)=.25$ Solve for $n$. My ...
3
votes
0answers
201 views

Maximum size of a rotated-then-cropped rectangle

With regard to topic/question New size of a rotated-then-cropped rectangle: The answer by Isaac, the maximum area is $b^2\csc\alpha\sec\alpha$ when $x=0.5b\csc\alpha = 0.5b/\sin\alpha$ seems to ...
4
votes
2answers
1k views

Simplifying a summation - stuck after change of summation index

I'm tracing through solution for a question I was working on. I don't quite understand how they got to the line which I marked with an arrow (apologies for using an image, I don't have much time left ...
0
votes
4answers
327 views

Finding the last two digits of the expansion of $2^{12n}-6^{4n}$

The question is: Find the last two digits of the expansion of $2^{12n}-6^{4n}$ where $n$ is any positive integer. If we put the value of $n=1$ we would get $2800$. For $n = 2$ the result will ...
1
vote
4answers
91 views

Range of functions on $\mathbb{R}$

While practicing problems on functions, I am getting a lot of incorrect answers from the given answers. I am a little confused by this as the problems seem simple else I haven't really understood the ...
3
votes
7answers
2k views

Effect of adding a constant to both Numerator and Denominator

I was reading a text book and came across the following: If a ratio $a/b$ is given such that $a \gt b$, and given $x$ is a positive integer, then $$\frac{a+x}{b+x} ...
0
votes
1answer
381 views

Shopkeeper's problem

I was reading a text book and came across the following interesting problem, mostly faced by shopkeepers: If $x\%$ discount on an article is given on cash payment, find the percentage that should ...
1
vote
3answers
179 views

Calculating overall Profit/Loss

I was reading a text book and came across the following interesting short-cut: If two items are sold, each at $X$, one at a gain of $P\%$ and the other at a loss of $P\%$, then overall loss ...
4
votes
2answers
4k views

Fundamental operations of mathematics

I have read that addition, subtraction, multiplication, exponentiation and division are considered the fundamental operations in math. Has this notion been made precise? Ie. is an operation ...
4
votes
4answers
238 views

Integral question: $\displaystyle\int \frac{x^{n-2}}{(1 + x)^n} {\rm d}x$

How would one integrate the following? $$\int \frac{x^{n-2}}{(1 + x)^n} {\rm d}x~$$ where $n$ is a positive integer.
5
votes
3answers
870 views

Why are horizontal transformations of functions reversed?

While studying graph transformations I came across horizontal and vertical scale and translations of functions. I understand the ideas below. $f(x+a)$ - grouped with x, horizontal translation, ...
0
votes
3answers
166 views

Is there a way to combine these equations?

Is there a way to rewrite these two equations into one equation that would work in both scenarios? If $x > y$, then the equation is: $$z = \frac{x - y}{1 - y}.$$ If $x < y$, then the equation ...
3
votes
2answers
241 views

Programming Logic: - Splitting up Tasks Between Threads

I asked this question at stackoverflow and instead of addressing the math required in the problem, they wanted to talk about why setting up 5 threads is no good, or question my intentions. I just want ...
9
votes
1answer
385 views

Necessary and sufficient condition for the coefficients of a quartic

I need some help with the following problem. Many thanks in advance. Let $f(x) = x^2+px+q$ and $g(x) = x^2+rx+s$. Find an expression for $f(g(x))$ and hence find a necessary and sufficient ...
0
votes
2answers
233 views

Rational function, sequences, polynomials and roots of unity

Let $$f(x) = \sum_{n\geq 0} a_n x^n = \frac{P(x)}{(1-x)^d}$$ be a rational function. (a) Prove: There is a polynomial $P_2(x)$ so $$\sum\limits_{n\geq 0} a_{2_n} x^n = \frac{P_2(x)}{(1-x)^d}$$ (b) ...
3
votes
3answers
209 views

$\frac{\pi}{2} =\tan^{-1}(\infty)$

Using the result, $\tan^2{\alpha} - A \tan{\alpha} + 1 = 0~$, where A is a constant, prove that the two solutions to this equation (such that $0 \leq \alpha \leq \frac{\pi}{2}$) are complementary ...
1
vote
2answers
638 views

Graph of root of quadratic function

This is a problem new to me. I need some guidance on what I should study/understand to be able to solve such problems. Sketch the graph of the following absolute functions on $\mathbb{R}$ and ...
1
vote
2answers
115 views

How do I compute $2ab\cos C$ given $a,b,C$? Isn't an operator missing there?

I have an equation. For example the law of cosines, $c^2 = a^2 + b^2 - 2ab \cos C$ So I calculate it all and I get something like this: 2500 cos 130. I calculate the cos 130, and get -0.643 Now what? ...
4
votes
2answers
292 views

Length of a number

Is there a method which would allow me to precisely calculate length of the largest number which has been created during multiplying two numbers? What I mean is this: 1258 * 2569 Multiplying the ...
3
votes
3answers
117 views

Solving $ \left| \frac{-2x-6}{4} \right| \le 5$ for $x$

Say I have a statement like: $$ \left| \frac{-2x-6}{4} \right| \le 5. $$ And I want to find the closed interval form of $x$. i.e. I want to know what the maximum and minimum $x$ can be. How do I ...
3
votes
2answers
113 views

Simple algebra question

$$\dfrac{\qquad\dfrac{5p+10}{p^2-4}\qquad}{\dfrac{3p-6}{(p-2)^2}}$$ Im all confused about this question. Can someone go through it step by step. Can you please list when I can cancel numbers. ...
1
vote
3answers
1k views

Smooth transition between two lines (2d)

I have function that is defined as $$ Y = \frac{1}{15} x \longrightarrow {\rm if}\qquad 0 \leq x \leq 30 $$ $$ Y = \frac{1}{70} x + \frac{11}{7} \longrightarrow {\rm if}\qquad x > 30 $$ The ...
13
votes
4answers
791 views

Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
1
vote
4answers
275 views

Moving variables around

Sorry for the easy question in advance, I'm trying to remember my algebra and I've come across a problem I don't quite remember the answer to: I've gotten the original problem (trying to get the $y$ ...
4
votes
4answers
333 views

$\tan(\frac{\pi}{2}) = \infty~$?

Evaluate $\displaystyle \int^{\pi}_{0} \frac{dx}{5 + 4\cos{x}}$ by using the substitution $t = \tan{\frac{x}{2}}$ For the question above, by changing variables, the integral can be rewritten as ...
6
votes
1answer
293 views

All functions $\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + \frac{2x^2}{f(y)}\right)$

How can I find all continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}^+$ such that $$\frac{1}{f\left(y^2f(x)\right)} = \big(f(x)\big)^2\left(\frac{1}{f\left(x^2-y^2\right)} + ...
6
votes
4answers
344 views

Best tool for verifying algebraic manipulations

Can anyone advise a tool that is very convenient for verifying algebraic manipulations step by step as I work. I too often waste hours dealing with a wrong formula because I missed a sign somewhere. I ...
3
votes
2answers
2k views

How do I combine multiple sets of ratios in order to meet an overall “target” ratio? (word problem supplied to illustrate)

(I had no idea how to tag this, any help would be great) :) I am a programmer with little math experience beyond high school - this will be demonstrated by my explanation. I am developing an ...
3
votes
1answer
3k views

Interesting percentage calculation shortcut formulae

I came across the following shortcut formulae while going through a text book: ...
-2
votes
1answer
376 views

How to find most negative number [closed]

How to find most negative number amongst (2M-1), (-2M+7), (-3M+10) and (-M+3). If we put M = 1 we get all positive answers. But we need most negative number. Please tell me what more information ...
2
votes
2answers
3k views

Square root of simple binomial function

Let's say I have the following formula: $$\sqrt{a^2-2ab+b^2}=\sqrt{(a-b)^2}=\sqrt{(b-a)^2}$$ When do I know which one of the following I should use?: $$\sqrt{(a-b)^2}=a-b\qquad\text{ or }\qquad ...
1
vote
2answers
8k views

Conversion from binary fraction to octal

Can someone verify my results? Binary to Octal 1) 10101.11 is 25.75 2) 0.01101 is 0.6875 3) 10110110.001 is 266.125
8
votes
2answers
325 views

Convergent rational series: which ones remain rational?

Due to closure under addition, it is obvious that a finite sum of rationals is rational. The infinite ones, however (assuming they don't diverge), may remain rational, such as $\sum_{n \in \mathbb{N}} ...
3
votes
3answers
136 views

Understanding algebraic manipulation

If $a+b+c \neq 0 $ where $a,b$ and $c$ are three non-zero distinct integers, then find the value of: $$\frac{ab+ca}{a^2+ab+ca} + \frac{ab+cb}{b^2+ab+bc} + \frac{ac+cb}{c^2+ac+bc}$$ What confusing me ...
3
votes
2answers
130 views

Factoring Quantities Question

I was doing an exercise and ran into a problem with their use of factoring. Here is the problem specific to where the issue occurs: $$ \frac{(x^2 + 1)^{1/2} - x^2 (x^2 + 1)^{-1/2}}{x^2 + 1} = ...
4
votes
3answers
2k views

Rationalizing the denominator

So I feel stupid for asking this, but I can't figure this out. I haven't taken algebra for about 8 years, so doing this is kind of fuzzy. Just started Calc 1 and we're finding limits. $$\lim_{x \to ...
2
votes
1answer
485 views

Rules applying to nested absolute values

I'm trying to use some algebra get $||x-5|-10|<\epsilon$ into a more manageable form (I'd like it in terms of $0<|x+5|<\delta$) but I'm not sure where to begin. I don't really know the rules ...
3
votes
2answers
486 views

Looking for a good precalculus/algebra reference

I'm working my way through Calculus, 9th ed. by Larson and Edwards in independent study. The problem is that many of the exercises and examples use a lot of algebra tricks that are glossed over and ...
3
votes
2answers
336 views

Continued proportion implies $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2$

I am trying to find a tricky way to proof these: If $a,b,c,d$ are in continued proportion,prove that $$(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2 ,$$ This result could be extended to ...
2
votes
3answers
740 views

Trouble with absolute value in limit proof

As usual, I'm having trouble, not with the calculus, but the algebra. I'm using Calculus, 9th ed. by Larson and Edwards, which is somewhat known for racing through examples with little explanation of ...
3
votes
2answers
224 views

Please explain this definition of symmetry

I am reading an article "Differential Equations: Not just a bag of tricks" in the mathematics magazine. The author has given elementary examples of symmetry ($y=x^2$ symmetric about $y$ axis, $y=x^3$ ...
2
votes
1answer
138 views

Proof involving complex numbers

I need a little help with this proof. I haven't got any idea how to proceed. The pseudo clue in the question doesn't clarify much. The real numbers $r$ and $\theta$, where $r > 0$ and $-\pi ...
4
votes
4answers
1k views

Find the centre of a circle passing through a known point and tangential to two known lines

I am trying to find the centre and radius of a circle passing through a known point, and that is also tangential to two known lines. The only knowns are: Line 1 (x1,y1) (x2,y2) Line 2 (x3,y3) ...