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
58 views

Root of equation, solvability

I was trying to solve the following equation for t $$(P\cdot l \cdot \exp(-l\cdot t) + R \cdot l \cdot \exp(-l \cdot t))/t + (P \cdot \exp(-l \cdot t) + R \cdot (\exp(-l \cdot t) - 1))/t^2 = 0 $$ ...
0
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4answers
187 views

Multiple choice question on rates of change (or so I thought)

If I were to find the resistance of the component (see image below), I would either find the equation of the curve and use differentiation or I'd draw a tangent at $V_2$ and then find the reciprocal ...
1
vote
5answers
100 views

Proof of basic properties

Can anyone provide proof of properties such as: $$a(b+c) = ab+ac$$ $$(a+b)^2 = a^2+2ab+b^2$$ And exponent rules: $$a^n \cdot a^m = a^{n+m}$$ $$(a^n)^m = a^{n \cdot m}$$ For $a, b, c \in ...
1
vote
3answers
150 views

Explaining something to the half

I'm a private tutor in my free time, teaching some basic high school mathematics and I've often been asked: ''Why is something to the half equal to the root of that something?''. And I'm having ...
1
vote
3answers
160 views

Solution Set of $\sqrt{x+1}+\sqrt{x-1}=1$

If $x$ is real, then find the solution set of $\sqrt{x+1}+\sqrt{x-1}=1$.
0
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1answer
1k views

Convert standard form of hyperbola to function form

I understand the concept of converting an equation of a hyperbola from general form into standard form, however I need to do the opposite. The equation is the following: ...
3
votes
2answers
196 views

Proving Injectivity $x + \sin(x)$

I'm trying to prove injectivity of a particular function (without calculus), but I've come across a bit of a problem. The function is: $$f(x) = x+\sin(x)$$ I started by (abiding by common standards) ...
0
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2answers
129 views

Compound interest coumpounded n time per year formula. $A=P\left(1+\frac{r}{n}\right)^{nt}$ intuition behind it.

I know that the compound interest formula for the interest compounded annually is given by $$A=P(1+r)^t$$ I know the intuition behind it. But why the compound interest formula for the interest ...
0
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1answer
34 views

How to convert a value that belongs to a range to its equivalent in another range?

Let's say we have the range $[0, 1]$ and the value $0.7$ that belongs to that range, how can I convert that value to its equivalent in the range $[0.8, 1]$? (or any other arbitrary range) Could you ...
0
votes
2answers
45 views

How do you get from this to this formula?

I have the formula : $$3×4^{n-1}×1×\left({1\over 3}\right)^{n-1}$$ And I would like to know how to get to this one (which is equal) : $$3× \left({4\over 3}\right)^{n-1}$$ How can I do that ?
3
votes
1answer
65 views

Proving no polynomial $P(x)$ exists such that $P(a) = b$, $P(b) = c$, $P(c) = a$

If $P(x)$ is a polynomial with integer coefficients and $a, b ,c$ are three distinct integers, then show that it is impossible to have $P(a) = b$, $P(b) = c$, $P(c) = a$.
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2answers
37 views

Explicit functional relationship

I have an implicit relationship between dependent variable $y$ and independent variable $x \in \mathbb{R}$ which read as follows: $$ \frac{(y-1)^{\alpha + 1}}{y} = \exp{(\beta x)}$$ Here $\alpha, ...
0
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3answers
52 views

Finding $a,b$ with two equations

$$ 16a + 4b + 1=37 \\ 64a + 8b + 1=25 $$ I am having trouble find the values of $a$ and $b$ in the equation above.
1
vote
2answers
106 views

How do you simplify $ \frac{\tan \theta \cos \theta}{\sec \theta} $?

How do you solve/simplify this? I am having trouble solving for the correct answer. We did this in class and I am getting a different answer than what the teacher said it was. $$ \frac{\tan ...
0
votes
1answer
34 views

How to derive geometric mean

Suppose that $(1+r_g)^n = (1+r_1)(1+r_2)$, how should I derive the formula such that $r_g = (r_1r_2)^{1/n}$. I am trying to prove that $r_g = (r_1r_2)^{1/n}$.
1
vote
1answer
38 views

How is this step completed?

User Did, did this step in his answer to my previous question: $$\sum_{k=0}^n{n\choose k}(zp)^kq^{n-k}=(q+pz)^n.$$ How is it done? Is it simply an identity, or something more?
32
votes
13answers
9k views

Algebra: What allows us to do the same thing to both sides of an equation?

I understand that the expressions on both sides of an equal sign are the same entity, and I know that when you modify one side, the other must be changed because it is referring to the same thing. ...
1
vote
3answers
1k views

How can I find the following product? $ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$

How can I find the following product using elementary trigonometry? $$ \tan 20^\circ \cdot \tan 40^\circ \cdot \tan 80^\circ.$$ I have tried using a substitution, but nothing has worked.
0
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2answers
37 views

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$?

Given that $\log_2(x)=p$ and $\log_4(y)=q$, how do I evaluate $\log_x(4y)$? There were some other questions like this and I applied this formula to them $\log_a(xy) = \log_a(x)+\log_a(y)$. However, in ...
0
votes
0answers
27 views

How to limit the result of an operation to a specific range?

Let's say I have $a = 1 / b$, and the highest possible value for $b$ is $1$ and the lowest is $0.1$. Because of this, the result range of the operation is from $1$ to $10$. I want to turn that from ...
17
votes
3answers
438 views

$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ$

How can I find the following product using elementary trigonometry? $$ \tan 1^\circ \cdot \tan 2^\circ \cdot \tan 3^\circ \cdots \tan 89^\circ.$$ I have tried using a substitution, but nothing ...
4
votes
4answers
152 views

why do equations work and how do they relate to each other?

Ok, so I understand that an equation is something like 15 = 15 , and that the only criteria as far as I can tell for it being an equation is that both sides are equal to each other. I have a few ...
0
votes
1answer
576 views

8 less than triple a number is equal to -5 (I'm trying to find the unknown number)

It's a hard question... for my thinking! I've tried many solutions but haven't figured it out. (Let $M$ be the unknown number) I've tried: $M^3-8=-5$ Then $M^3-8+8=5+8$ Then I think the answer ...
0
votes
2answers
108 views

no. of ways to arrange $8$ sailors from which $3$ on one side and $2$ on other side.

(1) Out of $8$ sailors on a boat , $3$ can work only on one side and $2$ only on other side. Then the number of ways the sailors can be arranged on a boat , is (2) Passengers are to travel by a ...
2
votes
0answers
75 views

“Taxes and Option Prices” (question about Derivatives Markets by McDonald)

Thanks in advance for any help, and please tell me if there's anything I can do to make things clearer. I am having trouble understanding appendix 10.A to Derivatives Markets by Robert L. McDonald. ...
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vote
2answers
50 views

Remainder theorem question

If $n$ is an integer, what is the remainder when $$3x^{2n+3}-4x^{2n+2}+5x^{2n+1}-8$$ is divided by $x+1$? How would we know what the value of $n$ is?
2
votes
1answer
74 views

How do you solve for $x$ in this equation?

$$ \frac{100}{9} = \frac{1 - \frac{1}{(1+x)^{12}}}{x} $$ I tried and tried, but can't seem to get $x$ into a form to isolate it or use a quadratic formula or imaginary numbers or something. I need ...
3
votes
1answer
96 views

How prove $((x-y)(y-z)(z-x))^2\le 2((x^2-y^2)^2+(y^2-z^2)^2+(z^2-x^2)^2)$

let $x,y,z>0$, and such $$x^2+y^2+z^2=x^2y^2+y^2z^2+x^2z^2$$ show that $$((x-y)(y-z)(z-x))^2\le 2((x^2-y^2)^2+(y^2-z^2)^2+(z^2-x^2)^2)$$ My try: let $$x-y=a,y-z=b,z-x=c\Longrightarrow a+b+c=0$$ ...
0
votes
2answers
206 views

How to rearrange this equation to solve for 'r' in closed form?

I'm taking a finance course, and I can't afford the financial calculator which can be used to solve this, so I would like to know how to solve this algebraically by hand (I don't care if it uses ...
2
votes
1answer
8k views

Do you use degrees or radians for trig functions?

I was just wondering if you use degrees or radians in trig functions. For example if I have a degree of 0.5 would I do: Sin(0.5) or would I have to convert that to radians? Or does it not matter ...
0
votes
1answer
79 views

An algebra/linear algebra question

Suppose 8 real numbers $a,b,c,d$ and $x,y,z,w$ satisfy \begin{equation*} a^2+b^2+c^2+d^2=x^2+y^2+z^2+w^2=1,\quad ax+by+cz+dw=0. \end{equation*} Is it true that \begin{equation*} ...
0
votes
1answer
1k views

Question about transform on digits

I need help with a problem. Forvany positive integer m, let u be its units digit and t be it's number to tens, that is, the integer obtained when u is removed from m. So m=10t+u. For any positive ...
6
votes
4answers
165 views

Prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's.

This was taken from an old Brazilian Mathematical Olympiad (1992). As the title says, we're supposed to prove that there is an integer $n$ such that $n^{1992}$ starts with $1992$ one's (in the ...
1
vote
4answers
347 views

Simple question about the range of possible values for a function

So we have $2 |3-x| + 5 = k$, where $k$ is a constant. Provided this equation has two real solutions for $x$, what is the range of possible values for $k$?
0
votes
3answers
74 views

Difficult equation question

I have the equation $ t\sin (t^2) = 0.22984$. I solved this with a graphing calculator, but is there any way to solve for $ t$ without graphing? Thanks!
1
vote
3answers
43 views

Solve $\sin^2 x-1=0$ for $0^\circ\le x\le360^\circ$

I am new to this so I don't know how to type the exponent. Basically, I have to solve for $x$ for the following equation $$\sin^2 x-1=0.$$
0
votes
2answers
58 views

Problem with root equation

Please help me solve this equation: $3\sqrt{2+x}-6\sqrt{2-x}+4\sqrt{4-x^2}=10-3x$ where x is a real number. Thanks.
5
votes
1answer
336 views

Question on Algebra

Let $P(x)$ be a fourth degree polynomial with coefficient of leading term be $1$ and $P(1) = P(2) = P(3) = 0$, then find the value of $P(0) + P(4)$ .
0
votes
2answers
44 views

Probability of three non-disjoint events(likely trivial)

This should be a trivial problem at this point, so it is embarrassing to ask, but here it goes: If I have the probabilities(of just these regions of a venn diagram, given by): ...
1
vote
1answer
71 views

Help finding equation of circle given another circle

Please help me find the standard equation of the circle passing through the point $(-3,1)$ and containing the points of intersection of the circles $$ x^2 + y^2 + 5x = 1 $$ and $$ x^2 + y^2 + y = 7 ...
0
votes
4answers
143 views

Grade 10: Maxima and Minima (Application Question)

This is my Question: A piece of wire 40cm long is to be cut into two pieces which are each bent into the shape of a square. Find the length of each piece of wire if the sum of the areas of the squares ...
0
votes
1answer
425 views

Find angle in radians on a Ferris Wheel

John has been hired to design an exciting carnival ride. Tiff, the carnival owner, has decided to create the world's greatest ferris wheel. Tiff isn't into math; she simply has a vision and has told ...
1
vote
3answers
82 views

Inequality involving conjugate numerator/denominator pairs

Question is to solve: $$\frac{(x-2)(x-4)(x-7)}{(x+2)(x+4)(x+7)} > 1$$ I thought I could negate terms to make them equal (i.e. $-(x-2)$), but that does not happen. I could subtract $1$ from ...
0
votes
1answer
70 views

Prove that if polynomial $f(x) = x^6 + ax^3 + bx^2 + cx + d$ null places are real, then$ a = b = c = d = 0$

if polynomial null places are all real, then how can i show that $a = b = c = d = 0$? $x(x(0x + 0 + x^4) + 0) + 0 => x^6 = f(x)$ ? But that doesn't prove anything for me i think
0
votes
1answer
348 views

Finding linear and angular speeds

(b) A wheel of radius 5 in. is rotating 45°/sec. What is the linear speed v, the angular speed in RPM and the angular speed in rad/sec? So for this one I thought the linear speed (v)= 5(pi/2) because ...
0
votes
1answer
34 views

Calculating value for alpha in ln

I have the following formula: $$ 2((\frac1\alpha)\space ln\frac{1}{(1-\alpha)} )= \frac{1}{1-\alpha} $$ After applying the ln rules and multiplying I end up with: $$ ...
0
votes
1answer
44 views

if $ -3\left(x-\lfloor x \rfloor \right)^2+2(x-\lfloor x \rfloor )+a^2=0$.has no integral solution, then $a$ is

if $a$ is real number and $\displaystyle -3\left(x-\lfloor x \rfloor \right)^2+2(x-\lfloor x \rfloor )+a^2=0$.has no real integral solution, Then all possible values of $a$ lie in the interval.. ...
0
votes
2answers
179 views

A hunting lodge has enough fuel to keep 20 room heated for fourteen days [closed]

A hunting lodge has enough fuel to keep 20 room heated for fourteen days . If the lodge decides to save fuel by turning off the heat in 5 unoccupied rooms, and each room requires the same amount of ...
3
votes
1answer
68 views

To solve $2^x+4^x+2^{\lfloor x \rfloor}+4^{\lfloor x \rfloor}+2^{x- \lfloor x \rfloor}-4^{x-\lfloor x \rfloor}=50+\sqrt{50}$

How to solve for positive real $x$: $$2^x+4^x+2^{\lfloor x \rfloor}+4^{\lfloor x \rfloor}+2^{x- \lfloor x \rfloor}-4^{x-\lfloor x \rfloor}=50+\sqrt{50}$$ ?
0
votes
3answers
77 views

Find primes $p_1,p_2,..,p_6$ such that $1+\prod_{i=1}^{6}p_i $is not prime

Show that if$$ p_1, p_2, p_3, p_4, p_5, p_6 $$are primes, then $$1+\prod_{i=1}^{6}p_i$$ is not necessarily prime by using a specic example.