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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-3
votes
2answers
170 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
438 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
182 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
327 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 ...
17
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
171 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
305 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 ...
-5
votes
4answers
133 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
442 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
597 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
364 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
712 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
271 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
119 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
91 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
230 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
170 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
106 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
247 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
207 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
92 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
279 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
688 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 ...
2
votes
4answers
98 views

how can I solve for $x$?

I am having a bit of trouble solving for x when trying to find $f^{-1}$. I have $$y=\frac{x+5}{x-4}$$ How can I get x on one side? I tried multiplying both sides by the denominator but I am still left ...
0
votes
1answer
93 views

Algebra help please, substitution method for these equations

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} ...
3
votes
4answers
275 views

solving for a coefficent term of factored polynomial.

Given: the coefficent of $x^2$ in the expansion $(1+2x+ax^2)(2-x)^6$ is 48, find the value of the constant a. I expanded it and got ...
5
votes
1answer
117 views

how does this simplify?

I am really stumped as to how it simplifies. I should know this, but it has been a very, very, long summer.
2
votes
1answer
155 views

How to constrain disks that intersection of them is inside unit circle

I have two disks $(x-a_1)^2+(y-b_1)^2\leq r_1^2$ and $(x-a_2)^2+(y-b_2)^2\leq r_2^2$, where $a_1$, $b_1$, $r_1$, $a_2$, $b_2$, $r_2$ are all known. What kind of constraint can I put on $a_i$, $b_i$ ...
3
votes
1answer
693 views

use an addition formula to derive the identity…

$\cos(x-\pi/2)=\sin x$ I understand I have to use $\cos(A+B)=\cos A\cos B-\sin A\sin B$. Am I wrong to let $A=x$ and $B= -\pi/2$? Can someone please explain to me in detail how to solve this? Thank ...
3
votes
4answers
246 views

Where is the order of the variables inside the parentheses coming from?

I'm reading Sawyer's Prelude to Mathematics, here: I can't understand what's the meaning and application of "condition" here. Also when he gives the example on the cubic equation, stating that ...
5
votes
3answers
143 views

How long will it take to fill this cistern?

Pipes $A$ and $B$ call fill a cistern in $20$ and $30$ minutes and C can empty it in $15$ minutes. If the three are opened and closed after the other successively for $1$ min each in that order, how ...
2
votes
1answer
122 views

Average price per person

I am having trouble judging where I am going wrong with the following question The price of lunch for 15 people was $207.00 including a 15% gratuity for service.What was the average price per ...
0
votes
0answers
95 views

How fast/slow in comparison with other methods for proving that certain integer (also finding square root) is odd perfect square in next method?

Example 1: $49 - (49-1)/4 = 49 - 12 = 37$ where $12 = 3 * 4$ and $37=4^3-3^3$ -centered hexagonal number and its square root is $3+4=7$ Example 2: $121 -(121-1)/4 = 121 - 30 = 91$ where $30 = 5 * 6$ ...
6
votes
4answers
199 views

Solve $x^2+x^2t^2+2xt^2+t^2=1$ for $x$.

Solving $$x^2+x^2t^2+2xt^2+t^2=1$$ for $x$ yields $$x=\frac{1-t^2}{t^2+1}.$$ I had to use software because I could not manipulate it algebraically, and sadly the program did not tell me how it reached ...
2
votes
2answers
115 views

general formula for :$ax^n+by^n$

If: \begin{eqnarray*} ax+by&=&m_1,\\ ax^2+by^2&=&m_2,\\ ax^3+by^3&=&m_3,\\ ax^4+by^4&=&m_4\\ &\vdots& \\ \end{eqnarray*} Where : $m_1,m_2,m_3,m_4$ are constants ...
2
votes
2answers
157 views

Evaluating $\sum_{k=1}^n k^x$ [duplicate]

Possible Duplicate: Finite Sum of Power? Is there a general expression for $\sum_{k=1}^n k^x$ for any integer value of $x$? The table for $x=1,2,\dots 10$ is given here. Is there formula ...
0
votes
3answers
350 views

How can one prove that $\lim\limits_{\theta\to 0} \frac{\sin\theta}{\theta}=1$ [duplicate]

Possible Duplicate: Proving $\lim_{\theta \to 0}{\frac{\sin \theta}{\theta}}=1$ using $\frac{1}{2}r^2(\theta-\sin \theta)$ How can one prove that $$\lim_{\theta \to 0} ...
2
votes
2answers
149 views

Exponentials: Don't understand the simplification step

Can you explain how they go from the first line to the second? More specifically, what are the steps for making the negative exponential terms positive? i.e. how does the third term: $-e^{j(4\pi t + ...
2
votes
4answers
297 views

Calculating the chess problem, sum $\sum_{k=0}^{63}2^{k}$ [duplicate]

Possible Duplicate: Summation equation for $2^{x-1}$ I'm solving the classic problem of the inventor of chess, who according to legend sold the invention for one grain for the first square ...
1
vote
2answers
69 views

What percent of Juan's income is Mary's income?

For the following question Mary's income is 60% more than Tim's income , and Tim's income is 40% less than Juan's income. What percent of Juan's income is Mary's income? (Ans=96%) I am ...