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
3answers
368 views

Solving a system of equations with natural numbers?

first of all I am sorry if the level of this question is nowhere near the usual level of questions on this site because my math knowledge is still very basic. I hope you won't mind. I found this ...
1
vote
1answer
369 views

How to get the sum/difference of functions

Can someone please explain how to solve this question: $f(x)=−3+2\cos(x)$ $g(x)=\cos(x−\pi/4)−2$ Sum of functions: $s(x)=f(x)+g(x) $ Difference of functions: $v(x)=f(x)−g(x)$ Get the sum and ...
1
vote
0answers
60 views

How to get the sum of functions

I'm having a tough time with this question: $f(x) = -3 + 2\cos(x) $ $g(x) = \cos(x- \frac{\pi}4) -2$ Sum of functions: $s(x) = f(x) + g(x)$ Difference of functions: $v(x) = f(x)-g(x)$ Get the sum ...
7
votes
1answer
334 views

infinity sum of numbers [closed]

If we have a series of numbers $$1^5 + 2^5 + 3^5 + \cdots + (10^n)^5.$$ Final sum of the series is approximately equal $16666\ldots$ . If there is more and more numbers in the series is the ...
0
votes
0answers
82 views

Eccentricity and Length of Semi Axes of a conic

If a conic $ax^2+by^2+2hxy+2gx+2fy+c=0$ and say: How to find the eccentricity and the semi-axes of this conic. I do understand that if its a hyperbola only one of the semi axes will be real. Soham
1
vote
1answer
798 views

How many different 20-person committees can be formed…

Please help me how to solve this question It's just confusing my mind. Q)How many different 20 persons committees can be formed each containing at least 2 professors and at least 3 Associate ...
1
vote
2answers
168 views

Three inequalities with sums of fractions over two positive integers

In a proof, I arrive at three inequalities for all $p,q \geqslant 0$: \begin{align} \frac{p+1}{q+1} + \frac{q+1}{p+1} &\geqslant 1 + \frac{p}{2q+1} + \frac{q}{2p+1} + \frac{1}{p+q+1};\cr ...
9
votes
3answers
918 views

Simplify $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$.

Denest $\sqrt {\sqrt[3]{5}-\sqrt[3]{4}}$. I have tried completing square by several method but all failed. Can anyone help me please? Thank you. p.s. I'm a poor question-tagger.
0
votes
5answers
1k views

How are different functions bounded if they have an asymptote?

Does a horizontal line count as bounded? Also, how is $y=2^x$ bounded?
4
votes
1answer
271 views

Upper and lower limits of $f(x)=\cos(x^2)-\cos((x+1)^2)$ as $x\to\infty$.

I believe that the upper limit is +2 and the lower limit is -2. We have the trigonometric identity $\cos(x^2)-\cos((x+1)^2)=2\sin(x^2+x+1/2)\sin(x+1/2)$. We then make the substitution $x=m2\sqrt{\pi}$ ...
1
vote
2answers
48 views

separable equations.. algebra?

So, I am failing to understand some potentially simple algebra here. I have a separable equation: $ \frac{dy}{dx} = \frac{e^{-x} - e^x}{3+4y} $ and after the easy integration $=>$ $3y + 2y^2 = ...
2
votes
2answers
178 views

Finding highest possible value of function

I am to determine the highest possible value of $p(t)$ when $$p(t) = 7t - 11 - 2t^2$$ I try completing the square: $$= -2\left(t^2 -\frac{7}{2}t + \frac{11}{2}\right)$$ $$= ...
2
votes
1answer
902 views

How to calculate minimum distance between two arbitrary ellipses in 2D?

Arbitrary ellipses means that they can be scaled, translated and rotated in any way in 2D. Do you know some high-school method (might be slightly more advanced than that) to find the minimum distance? ...
0
votes
0answers
59 views

Two bivariate rational functions

I have $2$ bivariate rational equations: $\dfrac{t(b_2^2-a_2^2-t^2b_2^2)+t^3a_2^2}{1+t^2}+\dfrac{ta_1a_2-ts^2a_1a_2-b_1b_2s+b_1b_2st^2}{1+s^2}=0$ ...
0
votes
1answer
283 views

Mathematics Word Problem (cannot work it out for the life of me)

First part of the question A bank was robbed this morning. A lone robber carried the loot away in a big leather bag. The manager of the bank said that the money stolen consisted of small bills: \$5, ...
0
votes
3answers
106 views

Calculating Operating Costs to give a fixed profit margin

We are importing sales data from a company we just purchased. We want to calculate the logistics value so that the profit margin is 13%. We have the following rules in our system: ...
5
votes
3answers
1k views

Factoring the expression $x^6 + 64$

Alright, so apparently I've factored this out wrong... $x^6 + 64 =$ $x^6 + 2^6$ Then I continued, using $a^3 + b^3 = (a + b)(a^2 - ab + b^2)$ to get ... $$(x^2)^3 + 64 = (x^2)^3 + 4^3 = (x^2 + ...
-3
votes
2answers
172 views

How would I simplify this expression?

$ \left(\dfrac{a^{2}b^{-{\color{red}3}}}{x^{-{\color{red}3}}y^{2}}\right)^{3}\left(\dfrac{x^{-2}b^{-1}}{a^{\frac{{\color{red}3}}{{\color{red}4}}}y^{\frac{1}{3}}}\right) $
1
vote
4answers
494 views

Work and Time calculation

A is thrice as good as a workman as $B$ and therefore is able to finish a job in $60$ days less than $B$. How much time will they take to finish the same job if they work together? My attempt: Let's ...
1
vote
1answer
191 views

The Way to solve an equation involving logarithms

How can I solve the following equation: $$2^{log_3 x}+x^{log_3 2}=4$$ I don't want the final answer, I want to know how I can solve these kind of equations.
1
vote
1answer
331 views

Find the value of $A$ and $B$ such that $P$ is a rational number

Well, I got stuck in that question: Consider the integral expression in $x$: $P = x^3+x^2+ax+1$ where $a$ is a rational number. At $a=A$ the value of $P$ is a rational number for any $x$ which ...
18
votes
5answers
2k views

How to solve $x^3=-1$?

How to solve $x^3=-1$? I got following: $x^3=-1$ $x=(-1)^{\frac{1}{3}}$ $x=\frac{(-1)^{\frac{1}{2}}}{(-1)^{\frac{1}{6}}}=\frac{i}{(-1)^{\frac{1}{6}}}$...
3
votes
2answers
180 views

The value of $\bigl\lfloor x\bigr\rfloor+\bigl\lfloor x^2\bigr\rfloor+\bigl\lfloor x^3\bigr\rfloor+\bigl\lfloor x^4\bigr\rfloor$

There was a multiple choices saying: Find the value of $\bigl\lfloor x\bigr\rfloor+\bigl\lfloor x^2\bigr\rfloor+\bigl\lfloor x^3\bigr\rfloor+\bigl\lfloor x^4\bigr\rfloor$ knowing that ...
7
votes
4answers
314 views

How would I solve $(x-3)(x-2)(x-1)\gt0$?

This problem is on my Calculus Readiness Test and I was having a lot of trouble with it. The problem is $$(x-3)(x-2)(x-1)\gt0$$ I know how to solve $(x-3)\gt0$ but I have never seen these type of ...
-4
votes
4answers
136 views

$\frac{2x + 10}{2} - x = 5$: How do I show that all kinds numbers satisfy this equation?

I want to show that this magic equation is true for all kinds numbers (rational, irrational, imaginary, natural, etc..) which is true as far as I've ...
2
votes
5answers
509 views

Converting an IF condition to a mathematical equation

I am trying to study about converting algorithms into mathematical equations. For this I just started with a simple random example : ...
2
votes
3answers
650 views

GRE Algebra question

A group can charter a particular aircraft at a fixed total cost. If 36 people charter the aircraft rather than 40 people, then the cost per person is greater by $12. (a) What is the fixed total cost ...
3
votes
2answers
1k views

Poor Score on Simple Currency Conversion Exam. What Happened? [closed]

So I just breezed (or thought I did) through a super routine currency conversion test. The score I received was a 6/10, though I cannot for the life of me figure out which questions I got wrong, other ...
2
votes
2answers
66 views

Solving a polynomial equation over a range of $x$?

$$\left.\left(\frac{2x^3}{3} - 4x^2 + 10x\right) \right|_1^3 = 12 - 20/3$$ I guess my first question would be what is the meaning of the line in this equation. I was under the impression it is meant ...
4
votes
5answers
373 views

Solve $\sqrt{x-4} + 10 = \sqrt{x+4}$

Solve: $$\sqrt{x-4} + 10 = \sqrt{x+4}$$ Little help here? >.<
3
votes
2answers
200 views

Can I prove this inequality algebraically?

With $x+y\ge z$ $(x,y,z\ge0)$, prove that: $$\frac{x}{1+x}+\frac{y}{1+y}\ge\frac{z}{1+z}$$ I'm aware that using analytic view this is easy since $f(x)=\frac{x}{1+x}$ is concave in $[0,\infty)$. ...
1
vote
4answers
823 views

Calculating the average daily revenue

I can't seem to solve the following problem any suggestions would be appreciated.. In a small snack shop the average revenue was $\$400$ a day over a $10$ day period. During this period, if the ...
4
votes
3answers
157 views

Expansion of $\bigl(\sum x_i\bigr)^4$

Show that $$(\sum_{i=1}^n X_i)^4=\sum_{i=1}^n X_i^4+4\sum_{i\neq j}^n X_i^3X_j+3\sum_{i\neq j}^n X_i^2X_j^2+6\sum_{i\neq j\neq k}^n X_i^2X_jX_k+\sum_{i\neq j\neq k\neq l}^n X_iX_jX_kX_l$$ Please show ...
0
votes
2answers
290 views

How can I find the range?

In the equation $f(x)=-x^2 +2x -7$, I am told the range is $(-\infty,-6]$. I feel the domain should be $(-\infty, \infty)$ because we can put any value in for $x$ and still get a real number. If the ...
1
vote
3answers
73 views

“Approximating” a Bernoulli process the wrong way

The larger a group of people is, the more probable it is that they have to wait for at least one person before they can finally leave for the next pub. Why? Assume the probability that one person ...
1
vote
3answers
90 views

How to solve for $ x$ when $x^2 \times a = (x-10)^2 \times b$

I'm trying to figure this one out, and I'm stuck. I'm going: $x^2 \times a = (x-10)^2 \times b $ $x^2 \times a/b = (x-10)^2 /b $ $a/b = \frac{(x-10)^2} { x^2} / x^2$ $\sqrt{a/b} = (x-10)/x$ ...
2
votes
2answers
120 views

What is the fastest or usual way to calculate $(\frac{x-1}{2})^2$ mod $x$ if $x$ is odd?

Because: A) for odd $x$ and $x \equiv 1\pmod {4}$ the upper formula is the same as $x - (x-1)/4$ B) for odd $x$ and $x \equiv 3\pmod {4}$ the upper formula is the same as $(x + 1)/4$ Example A) ...
2
votes
2answers
92 views

Need Help With Calc Homework Question

Suppose that $x^*$ means $5x^2 - x$. Then what does $(y+3)^*$ mean?
1
vote
1answer
37 views

Finding height from voltage

I'm using a 45" fuel rod which gives a reading of 0 - 5 volts depending on where the float sensor is on it. At the bottom the fuel rod reads 0, at the top it reads 5 and in the middle it reads 2.5. ...
3
votes
2answers
77 views

Prove that there is no value of the integers $x,y,z$ satisfied the equation: $19^x + 5^y + 1980z = 1975^{4^{30}} + 2010$

Prove that there is no value of the integers $x,y,z$ satisfying the equation: $$ 19^x + 5^y + 1980z = 1975^{4^{30}}+ 2010 $$ The equation $1975^{4^{30}}$ is like a double exponent :( thanks again, ...
3
votes
2answers
1k views

3 nonlinear equations in 3 variables - with the same variable raised to different powers

Hoping someone can give me some guidance on solving the following system of 3 nonlinear equations in 3 variables: $$\left\{\begin{align*} &x^2+y^2=100\\ &xy+yz=-102\\ &y^2+z^2=117 ...
1
vote
4answers
233 views

Is this a typo in my math book?

I am doing homework which is submitted online. I came across a question asking if two functions are equal. $f(x)=3x+4$ and $g(x)=14+(8/x)+b(x-4)$. I set the two equations together and got 7 for an ...
2
votes
1answer
177 views

Algebra Word Problem

If I drove 38.91 miles, and my car gets 22 miles to the gallon, and gas cost $3.699 per gallon, how much did I spend on gas? I'm sure this is an algebra problem, but I can't think where to put the X ...
3
votes
2answers
108 views

Two solutions for $x$

I am trying to solve this. I already got the answer but my doubt is if I am doing it right. There are two solutions for $x$ in the equation, to 2 decimal places. What is the value of the greater ...
6
votes
3answers
302 views

What do three consecutive dots mean in algebra?

In the formula, $$\frac{n(n-1)(n-2)\cdots(n-r+2)}{(r-1)!}a^{n-r+1}b^{r-1}$$ what does the "$\cdots$" mean?
0
votes
3answers
221 views

Smallest value of image set of a function

I have this equation where the question says, What is the smallest value, to 2 decimal places, in the image set of the function $f(x)=ax^2+bx+c$ ...
3
votes
1answer
94 views

Find whether or not an inverse exists algebraically

Is there an algebraic(without graphs) way to determine the existence of a function's inverse without using calculus? I'm an undergrad engineer and can obviously solve this using basic calculus, but ...
1
vote
2answers
320 views

Power curve calculation

I have a power curve relating to a turbo trainer I use on my bike. I save my sessions on a website called strava using speed, cadence, heart rate and time. Using the power curve I've been given I ...
2
votes
3answers
129 views

Little help with some algebra

Note: This isn't homework, I'm skipping ahead of class. Please answer all these equations, I'm deathly stuck on them. Use the substitution method only please. (Find $x$ and $y$.) \begin{align} ...
2
votes
3answers
727 views

How was the quadratic formula found and proven? [duplicate]

Possible Duplicate: Why can ALL quadratic equations be solved by the quadratic formula? History of Quadratic Formula How was the quadratic formula $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ found ...