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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0
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1answer
34 views

Solve this system of equations without calculator

$$2a +4b +3c +5d +6e=37$$ $$4a +8b +7c +5d +2e=74$$ $$-2a -4b +3c +4d -5e=20$$ $$a +2b +2c -d +2e=26$$ $$5a -10b +4c +6d +4e=24$$ find $a,b,c,d,e$ I tried solving the system of equations above but ...
0
votes
2answers
31 views

What happens to the graph of $f$?

I'm trying to figure out what happens to the graph of $f$ in the following to situations: $f = f(|x|)$ $f = f(\frac{1}{x})$ For the first, I know if $f = |f(x)|$ then the points below the $x$ ...
1
vote
1answer
25 views

Finding the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$

If $x,y \in \mathbb {R}$, find the minimum of $x^2+y^2$ when $(x^2y-xy^2)(x^3-y^3)=x^3+y^3$ and $xy>0$. This problem was inspired by a problem which asked if $x,y \in \mathbb {R}$ and $xy \neq ...
3
votes
1answer
13 views

Choosing a combination of books, under given restrictions.

Mary has on her bookshelf 5 novels, 5 biographies, and 8 textbooks. Mary decides to take three novels and four non-fiction books with at least one of the non-fiction books a biography. How many ...
0
votes
2answers
35 views

Why is $\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$ [on hold]

Why is this true? $$\cos\left(\frac{3\pi}{2}-t+2k\pi\right) = -\sin(t)$$
-1
votes
0answers
21 views

Find the positive integers $\overline {abc}$ such that $\frac{1}{a} +\frac{1}{b}+\frac{1}{c}$=$\frac{\overline {1b}}{\overline {1a}}$ [on hold]

Find the positive integers $\overline {abc}$ such that $\frac{1}{a} +\frac{1}{b}+\frac{1}{c}$=$\frac{\overline {1b}}{\overline {1a}}$.Can you help me with one solution without to consider the ...
0
votes
1answer
10 views

Write the particular equation expressing cost in terms of miles traveled

To take a taxi in downtown St. Louis, it will cost you $3.00$ to go a mile. After $6$ miles, it will cost $5.25$. The cost varies linearly with the distance traveled.
2
votes
0answers
34 views

Expanding trigonometric functions with binomial expansion

I was challenged to take $\cos^{\pi}(\pi)$ and expand it using binomial expansion and $\cos(x)=\frac{e^{xi}+e^{-xi}}2$, which I tried: $$\cos^{\pi}(\pi)=\left(\frac{e^{\pi i}+e^{-\pi ...
0
votes
3answers
52 views

Where is the mistake in solving the inequality?

Where am I going wrong in solving this inequality? $$\frac{p-\sqrt{9p-20}}{p-5}<2$$ On cross multiplying and squaring to remove the square root,I get the inequation $p^2-29p+120<0$ Which ...
3
votes
3answers
300 views

Find root of the equation

Find maximum root of the equation $$x - \frac{1000}{\log 2} \log x = 0$$ It locates between $13746$ and $13747$, but I want to find right solution not using graphing calculators. Thanks in advance.
1
vote
2answers
21 views

Condition for roots to lie in certain intervals

The set of values of $p$ such that both the roots of the equation $$f(x)=(p−5)x^2−2px+(p−4)=0$$ are positive and one of the roots is less than $2$ and the other root lies between $2$ & $3$ ...
0
votes
5answers
38 views

Equation $\log(x^2+2ax)=\log(4x-4a-13)$ has only one solution; then exhaustive set of values of $a$ is

Equation: $$\log(x^2+2ax)=\log(4x-4a-13)$$ It has only one solution; then exhaustive set of values of $a$ is ?? I don't even know where to begin The answer is : $$(-13/4,-13/12) \cup [-1]$$
1
vote
1answer
45 views

Prove ${20n \choose 10n}\ge {2n-1 \choose n-1}^{10}$

As the title says, I can't prove that, no matter what I try. What I've tried so far: induction: seemed the most obvious method, since we already had a lot of tasks with it, but using the esimates ...
1
vote
3answers
27 views

For how many days will the food last in garrison?

A garrison has sufficient food for $75$ soldiers for a period of $90$ days. After $10$ days, one third of the soldiers leave. After another $10$ days, $5$ soldiers return, From this day on, ...
0
votes
1answer
36 views

Divisibility test for 720 [on hold]

Use the divisibility test where possible to list all factors of 720 Please show further examples where appropriate, thank you.
0
votes
1answer
22 views

Solving for numerator in equation with logarithms (Activation Energy Equation)

I'm having trouble solving for k1 in this equation: ln(0.286/k1) = (100000/8.314)(1/500 - 1/490) The right side should equal 0.491, which I can calculate just fine, but then the left side gives me ...
2
votes
1answer
32 views

Complex derivative numerically using real $h$ and imaginary $h i$?

I want to find numerically (the functional expression might become too complicated) the derivative of a complex function (to use it in a Newton-algorithm). Can I simply do something like $$ \frac ...
0
votes
1answer
29 views

$x^J = y$, $J = 2.455\ldots$ What's the rest of $J$?

I have a problem where I need to know what J is. I do x^J and get y. For example, if I do 5^J, I would want to get 55 as y. Same with 4^J = 30. When J is 2.455, it works up to 4 only! I need for ...
2
votes
2answers
39 views

Method for solving the equation $3 + 5x^{1/2} = 2x$?

As I'm not even sure what type of problem this is, so I can't research it. $$3+5x^{1/2}=2x$$ My Question: Could I get an explanation (with example) of how to solve the above equation? Or direct me ...
-3
votes
1answer
27 views

Notational problem [on hold]

Please, how do I write the following as a combination of a sum and a product: $$ (c-a_1)(c-a_2)(c-a_3)b_1 + (c-a_2)(c-a_3)b_2+(c-a_3)b_3 ?$$ Also, how can I generalize it?
2
votes
1answer
33 views

Are these two events $A$ and $B$ independent?

Abe and Bernard are dealt five cards each from the same $52$ card deck. Let $A$ be the event that Abe gets a flush (five cards of the same suit) and $B$ be the event that Bernard’s five cards are of ...
4
votes
1answer
25 views

$p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent, then $p(X \wedge Y \wedge Z) > 0$?

Is the following statement true or not? Let $X$, $Y$, $Z$ be $3$ events in the same sample space such that $p(X)$, $P(Y)$, $p(Z) > 0$ and every pair of these events is independent. Then $p(X ...
1
vote
1answer
75 views

Why does $\sum\limits_{n=0}^{+\infty} z^n=\frac{1}{1-z}?$

Having $f(z)=\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n$ I had to find what $\sum\limits_{n=0}^{+\infty} \frac{1}{n!}z^n\sum\limits_{n=0}^{+\infty} \frac{D_n}{n!}z^n=\sum\limits_{n=0}^{+\infty} ...
1
vote
2answers
11 views

Find parallel line value

I've an homework problem that i'm unable to find the right answer. The problem is: The line $tx + sy = 2$ goes through point $(2,1)$ and is parallel to line $y = 8 -3x$, find the value of $t^2 + ...
0
votes
1answer
21 views

How do you calculate the change in thickness of a cylinder, if you shave off a flat section?

I have a piece of steel, cylindrical (hollow), 200mm outside diameter with 160mm inside diameter (...
7
votes
3answers
121 views

How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality?

Problem: How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality? My attempt: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
0
votes
1answer
24 views

What order do I write algebraic math problems?

My math teacher is a little bit picky on the order we write our simplified algebraic math problems, and I forgot to take notes on this. Let's say you have the problem $2(y + 3) - 4x$. Would I write ...
0
votes
2answers
27 views

How to solve a quadratic inequality that acts like a quadratic equality?

This will be largely a trivial question. But how do I solve an inequality like this: $3x^4 - 4x^2 + 1>0$ ? Of course, I can treat it like a quadratic inequality by saying $t=x^2$ So I can solve ...
0
votes
1answer
23 views

If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then the value of m is equal to to?

If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then the value of m is equal to to ? I drew the graph of $|x^2+4x+3|$.I found that that for the given condition $mx-2m$ must be ...
9
votes
6answers
785 views

Squaring both sides when units are different?

Given $((9) \text{inches})^{1/2} = ((0.25) \text{yards})^{1/2}$, then which of the following statements is true? $((3) \text{inches}) = ((0.5) \text{yards})$ $((9) \text{inches}) = ((1.5) ...
1
vote
1answer
46 views

If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$

If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$ and $\frac{\alpha}{\beta}$,$\frac{\beta}{\alpha}$ are the roots of $x^n+1+(x+1)^n=0$, then $n$ is Odd Even ...
-1
votes
0answers
20 views

Is $ \lfloor {\log(n)} \rfloor!$ or $ \lfloor {\log(\log(n))} \rfloor!$ polynomially bounded? [on hold]

Which of these is is polynomially bounded: $ \lfloor {\log(n)} \rfloor!$ $ \lfloor {\log(\log(n))} \rfloor!$ I think both are but I can't prove it.
0
votes
0answers
5 views

Is it possible to work out each one of these variables if it is the only unknown? $d=c_1c_2\ln(\cosh(t/c_2))$

I have an equation that defines $d$: $$d=c_1c_2\ln(\cosh(t/c_2))$$ It is very simple to work out $c_1$ if it is the only unknown: $$c_1={d\over c_2\ln(\cosh(t/c_2))}$$ Each variable is a real ...
2
votes
2answers
32 views

How to express this expression in terms of N?

I am trying to express this formula in terms of N: $$ A=\frac{a^2N} {a\cdot \tan{\frac{180}{N}}} $$ I really don't know how to do this. I tried and got this: $$ A \cdot a \cdot \tan{\frac{180}{N}} ...
1
vote
0answers
59 views

Show that the equation $x^2+y^2+z^2= (x-y)(y-z)(z-x)$ has infinitely many solutions in integers $x, y, z$.

Show that the equation $x^2+y^2+z^2= (x-y)(y-z)(z-x)$ has in finitely many solutions in integers $x, y, z$. It seems like if I find a set of $x,y,z$ that satisfy this for any values that will ...
-1
votes
2answers
34 views

Triangle with a square in it, with the side of $2\sqrt{3}$, what's the altitude of the triangle? [on hold]

We have a triangle and within is a square with the side of $2\sqrt{3}$. What's the altitude of the triangle ABC? All 3 angles in the triangle are same (60). Pic: http://imgur.com/gallery/SAmhU7z/new ...
3
votes
1answer
23 views

If $z_0$ is a root of the equation $z^n\cos\theta_0+z^{n-1}\cos\theta_1+\cdots+\cos\theta_n=2$

If $z_0$ is a root of the equation $z^n\cos\theta_0+z^{n-1}\cos\theta_1+\cdots+\cos\theta_n=2$, then $|z_0|<1/2$ $|z_0|>1/2$ $|z_0|=1/2$ Using triangle ...
0
votes
1answer
32 views

How to find radius of hemisphere in applied problem

If the stem of a mushroom is modeled as a right circular cylinder with diameter $1$, height $2$, its cap modeled as a hemisphere of radius $a$ the mushroom has axial symmetry, is of uniform ...
1
vote
3answers
38 views

Proof of divisibility: $17 \mid 3 \cdot 5^{2015} + 2^{2017} \cdot 5^{670}$ [on hold]

As the title says, prove that $3 \cdot 5^{2015} + 2^{2017} \cdot 5^{670}$ is divisible by $17$.
2
votes
2answers
34 views

Calculate $\sqrt{x^2+y^2+2x-4y+5} + \sqrt{x^2+y^2-6x+8y+25}$, if $3x+2y-1=0$

As the title says, given $x,y \in \mathbb{R}$ where $3x+2y-1=0$ and $x \in [-1, 3]$, calculate $A = \sqrt{x^2+y^2+2x-4y+5} + \sqrt{x^2+y^2-6x+8y+25}$. I tried using the given condition to reduce the ...
2
votes
2answers
23 views

Divisibility: $60 \mid (2x-y)(2y-z)(3z+2x)$, if $8x-10y+27z=0$

As the title says, given $x,y,z \in \mathbb{Z}$, where $8x-10y+27z=0$, prove that $(2x-y)(2y-z)(3z+2x)$ is divisible by $60$. I tried to bring the formula in a format of $(\cdots)(8x-10y+27z) + ...
2
votes
1answer
47 views

Find all functions $F(x)$ for which $F (x) + F ((x − 1)/x) = 1 + x$

Let $F (x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equation $F (x) + F ((x − 1)/x) = 1 + x$. Find $F (x)$. This ...
3
votes
3answers
55 views

Basic algebra problem: $ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $

Basic algebra problem I can't seem to figure out: $$ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $$ $x,y \in \mathbb{R}, x^2 \neq y^2, xy\neq0$. Now I know the result is: ...
1
vote
2answers
38 views

Probably very basic Euclidean geometry; Why is the following expression valid for a point along a straight line?

I am looking at constructible points in abstract algebra, particularly in $\mathbb{C}$. Alongside a proof of a theorem, I came across this expression which I cannot work out how it's been derived. It ...
1
vote
3answers
47 views

How to simplify this equation to solve for m?

It has been way too many years since high school. How can I simplify this equation to solve for m: $\frac{x}{c+pm}=m$ I got to $x = cm + pm^2$ and I don't know how to get any further. I wish this ...
0
votes
0answers
32 views

Squaring the square root of $x$ vs the square root of $x^2$ [duplicate]

While reviewing this algebra concept yesterday I realized I could not justify why this $\sqrt{x^2}=|x|$ is different from this $(\sqrt{x})^2=x$. My confusion stems from the fact that radicals can be ...
-3
votes
5answers
92 views

Catherine is now twice as old as Jason but 6 years ago she was 5 times as old as he was. How old is Catherine now? [on hold]

This is an IQ question. "Catherine is now twice as old as Jason but 6 years ago she was 5 times as old as he was. How old is Catherine now?" How to solve such questions? I think their combined age ...
2
votes
3answers
39 views

Factorise Algebraic Expression

Background: I came across the following problem in class and my teacher was unable to help. The problem was factorise $x^6 - 1$, if you used the difference of 2 squares then used the sum and ...
0
votes
2answers
32 views

How to prove a solution of equation is rational if another one is rational number?

The question is : $r$ is the solution of equation $x^2+bx+c=0$ and $r$ is a rational number, so there is another solution $s$, how to prove s is a rational number as well? I have no idea about it and ...
0
votes
0answers
20 views

Logarithmic function transformations

The standard log function form is $a \log[k(x-d)] + c$ Where $a$ vertically stretches or compresses $k$ horizontally stretches or compresses $d$ translates left or right $c$ translates up or ...