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
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1answer
27 views

How to find a corner by two points?

I've got an axis that contains 3 points. which one of them is in the center, and the other are in their corners. I have the position of the B and C, and i need to find A. How can i do that? That's a ...
1
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0answers
34 views

Coefficients of rational involution.

Question: (Spivak Calulus 3rd, Chapter 3, Problem 8) For which numbers $a,b,c,d$ will the function $f(x) = \frac{ax + d}{cx + b}$ satisfy $f(f(x)) = x$ for all $x$? Attempt at an answer: I tried ...
1
vote
0answers
16 views

How an I create a model population?

I need to crate a model population for my Math fair and this is what I have. I need some help with it -I want to find out the size of a population by using a natural exponential function -When I ...
0
votes
1answer
35 views

Calculating variable in math equation

I am not good with math, I have this equation (very simple to most) but I need help on how to get the value of x 10 = x - (1.29 + 4.99% of x) my question is how ...
1
vote
2answers
326 views

How can I prove this two identities? $\cos^2 x=\frac{1+\cos(2x)}{2}$ and $\sin^2 x=\frac{1-\cos(2x)}{2}$

How can I simply prove the two following equations? $$\cos^2x=\frac{1+\cos(2x)}{2} \,\,\,\,\,\,\,\,\,\,\text{ and }\,\,\,\,\,\,\,\,\,\, \sin^2 x=\frac{1-\cos(2x)}{2}$$ I already proven them using ...
0
votes
1answer
31 views

absolute value in a quadratic

If $a<-2$ is a real number, then the equation: $x^2+a|x|+1=0$ has how many real roots? After finding the roots in terms of $a$, how do I proceed?
6
votes
3answers
140 views

Find all real solutions for $16^{x^2 + y} + 16^{x + y^2} = 1$

Find all $x, y \in \mathbb{R}$ such that: $$16^{x^2 + y} + 16^{x + y^2} = 1$$ The first obvious approach was to take the log base $16$ of both sides: $$\log_{16}(16^{x^2 + y} + 16^{x + y^2}) = 0$$ ...
1
vote
1answer
209 views

Simplification of $\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}}$

I'm having trouble understanding how this expression: $$\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}} \cdot ...
1
vote
1answer
93 views

How do I prove that $\cos(\frac{1}{2}x)$ is a periodic function?

Given: $f(x)=\cos(\frac{1}{2}x)$. Prove: f is a periodic function with period 4π My math teacher never went over this so I don't know where to start or what to do :/
0
votes
1answer
129 views

Simplifying Sigma Notation

I am working on the proof on showing the ratio of two consecutive Fibonacci numbers converges to the golden ratio to explain to a student I am tutoring. I am getting to some confusion in a ...
0
votes
1answer
112 views

Are Parabolas similar intuitively?

All parabolas are similar, but are they all similar in that it is just a question of 'zooming in and out' intuitively speaking? It seems that there should therefore be on all parabolas a curve from ...
3
votes
1answer
88 views

Finding $n \in \mathbb{Z}$ such that $\sqrt{(4n-2)/(n+5)}$ is rational.

I have to find $n \in \mathbb{Z}$ such that $$\sqrt{\frac{4n-2}{n+5}}\in\mathbb{Q}.$$ I've expressed $\sqrt{\frac{4n-2}{n+5}}$ as $\sqrt{4 - \frac{22}{n+5}}$ and I think that there's no $n$ for it ...
0
votes
2answers
142 views

Finding the quadratic coefficient of a quartic polynomial given other coefficients.

This is a question which I want to solve, taken from this sample question paper for an exam I'm appearing for tommorow: The product of two of the roots of $$x^4-11x^3+kx^2+269x-2001$$ is $-69$. ...
1
vote
1answer
105 views

How to apply the remainder theorem for multi-variable polynomials?

In our math class, we were taught that for a polynomial $f(x)$ $$f(\alpha) \equiv f(x) \pmod {x-\alpha} $$ That's all very well. But, what about polynomials in more than variable? Specifically, how ...
2
votes
1answer
102 views

Calculate Interior Dimensions Using Cubic Feet

I have been tasked to calculate the interior dimensions of a product given some non-traditional values. Given the following numerical information how would one go about calculating the interior ...
8
votes
3answers
694 views

Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$

Calculate all integer solutions $(x,y,z)\in\mathbb{Z}^3$ of the equation $x^2+y^2+z^2 = 2xyz$. My Attempt: We will calculate for $x,y,z>0$. Then, using the AM-GM Inequality, we have $$ ...
41
votes
15answers
14k views

Why can't you square both sides of an equation?

Why can't you square both sides of an equation? I've been asked this many times and can never quite give a good, clear, concise answer (for beginning algebra students) in plain language. I just ...
0
votes
1answer
55 views

help on manipulating this algebraic expression

So I have something like: $\frac {k!}{(k-3)!3!}$ I'm going to add $\frac 12k(k-1)$ to this, and I want to obtain $\frac {(k+1)!}{(k-2)!3!}$ as the result. I'm having trouble with this since I need ...
0
votes
1answer
556 views

How to rearrange this equation to find the value of $L$

This question has been difficult for me to answer for awhile. If anyone could help that would be great. The period (the time for one complete swing back and forth) $p$, in seconds of a pendulum is ...
0
votes
2answers
146 views

Find the equation defining a perpendicular bisector

Hello fellows, I've not had much time to post questions, but I post this one because while in my Maths lesson, I became annoyed by solving the same thing over and over again, when a good ...
1
vote
1answer
57 views

How do I simplify this exponential expression?: $ 2(-3x^{-2}y^3)^{-1} \cdot (-3x^{-3}\cdot y)^2 $

How do I simplify this expression? Simplify: $$ 2(-3x^{-2}y^3)^{-1} \cdot (-3x^{-3}\cdot y)^2 $$ I tried and didn't get the answer.
2
votes
1answer
130 views

Markup reverse to get to original amount?

I am trying to figure out how to mark something up then reverse it. I am selling an item for $100$ and I need to cover a $15\%$ advertising expense. I want to net $100$ after advertising is paid for. ...
1
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3answers
72 views

Proving that if these quadratics are equal for some $\alpha$, then their coefficients are equal

Let $$P_1(x) = ax^2 -bx - c \tag{1}$$ $$P_2(x) = bx^2 -cx -a \tag{2}$$ $$P_3(x) = cx^2 -ax - b \tag{3}$$ Suppose there exists a real $\alpha$ such that $$P_1(\alpha) = P_2(\alpha) = ...
4
votes
3answers
200 views

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$.

Find all polynomials $P(x)$ such that $2xP(x)=(x+1)P(x-1)+(x-1)P(x+1)$. Well, if $\deg P\le 3$ this is easy since we can deduce $P(0)=P(1)=P(-1)$ by letting $x=0,1,-1$
1
vote
1answer
75 views

Find the smallest value for n for which $a_n ≥ 1,000,000$

The recursive formula is: $a_n = (a_{n-1})^2$, with $a_1 = 2$. I got that $a_5$ is the smallest value before it is equal to or greater than $1,000,000$. If I'm correct $a_5$ is $65,536$. I got that ...
2
votes
1answer
2k views

Determine the cube roots of -8 in polar form

Exam time tomorrow and I am not entirely sure if I am doing this right. I first write -8 as a complex number $z^3 = -8 = -8 \times 0i$ Calculate the modulus of z $|z| = \sqrt{-8^2} = 8$ Get the arg ...
1
vote
1answer
724 views

Getting rid of the square roots in the expression $\sqrt{a} + \sqrt{b} + \sqrt{c} = d$

I need to find an alternate form of $\sqrt{a} + \sqrt{b} + \sqrt{c} = d$ without square roots for a problem that I'm working on, but it's rather complicated to do. What we can do is $\sqrt{a} + ...
0
votes
3answers
185 views

Why does $0,\bar{9}$ equal $1$? [duplicate]

I am finding hard to understand why $0,99999..... = 1$ I have the following proof: Let $x$ be $0,9999...$ then $10x = 9,999...$ So $10x - x = 9,999 - 0,9999$ $9x = 9 \rightarrow x = 1$ From a ...
1
vote
3answers
291 views

How does one use dimensional analysis to check for errors?

In this post: http://math.stackexchange.com/a/438618/66675 , Ross Millikan explains how one can use dimensional analysis to check for errors even if the equation itself has no units. Can someone ...
-1
votes
2answers
237 views

Prove Why is Less than a number equals to this?

I have this word problem 68 less than 5 times a number is equal to the number. Find the number. my answer was $68 - 5x$ but the correct answer was $5x-68$. ...
0
votes
1answer
355 views

Average score of a batsman's innings

A batsman's runs just before the last match of the season, adds up to $750.$ In his last $2$ innings, he scores only $6$ runs, and his average drops by $2.$ Find his final average of the season. ...
1
vote
1answer
59 views

Maxima/minima of $f(x)=\frac{\sin(\frac{1}{2} Nx) }{\sin(\frac{1}{2} x)}.$

How do I find: the $\bf maxima$ and minima of the function $f$ with $ f$ given by: $$f(x)=\frac{\sin(\frac{1}{2} Nx) }{\sin(\frac{1}{2} x)}, \;\;(N=1,2,3...)$$ What I did, is: Minima: I set: ...
2
votes
2answers
43 views

Technique to solve this equation of 2 unkowns

I was solving a problem of single phase eletrical circuits where I had to find the inductor $L$ and resistance $R$. I managed to get two equations containing the two unknowns. ...
4
votes
1answer
241 views

Average of 3 consecutive odd numbers

The average of $3$ consecutive odd numbers is $14$ more than one third of the first of these numbers, what is the last of these numbers? $17/19/15/$data inadequate/none of these Let three ...
0
votes
1answer
2k views

variable with negative exponent in the denominator moved to nominator and vice versa

The top and bottom of the fraction both contain negative exponents. Since $c^{-3}$ on the bottom has a negative exponent, it is moved to the top of the fraction (numerator). Since the $d^{-3}$ on the ...
0
votes
1answer
101 views

derive a parabola from two tangent lines

I have two tangents of a parabola, one representing initial speed (y=mx) and the other the maximum permissible extent (y=e), and I want to find the gentlest deceleration required to stop in time. It ...
1
vote
1answer
229 views

Solve $x - 2\arctan(x)= 0$

$x - 2\arctan(x)= 0$. I can find one root (0) from the equation $\tan(x/2) = x$ but there are two others, namely ($-2.3312, 2.3312$) that I don't know how to find. Looking for help! Thanks :) ...
1
vote
2answers
138 views

Confusion regarding the Logarithmic function change of base formula

My textbook seems to be making a big leap when trying to prove the change of base formula for logarithms. If someone could help clear this up it would be very appreciated. It starts with: $b^{x ...
1
vote
1answer
261 views

How to prove this equals something else.

I'm trying to prove that $$\dfrac{l}{c+u}+\dfrac{l}{c-u}=\dfrac{2l}{\sqrt{c^2-u^2}}$$ My workings so far: $$\dfrac{l}{c+u}+\dfrac{l}{c-u},$$ put over common denominator, ...
0
votes
3answers
80 views

Finding a formula for the sequence $10, \,110,\,1110,\,11110,\dotsc$

How can I find a formula for the sequence $$10,\,110,\,1110,\, 11110,\dotsc$$ to make it ready for summation?
0
votes
2answers
135 views

Solve discrete Math Problem using abstract algebra, postage problem?

The question I am looking at is not very hard: Determine which amounts of postage can be written with $5$ and $6$ cent stamps. To determine the amount, use a brute force way to solve it. Counting ...
2
votes
2answers
1k views

How to prove that $\lim_{x \to 0} \sin(x) = 0$ using the epsilon-delta definition?

How do I prove that $\lim_{x \to 0} \sin(x) = 0$ using the episilon-delta definition of a limit? Do I have to divide the domain of $x$ into 4 cases for each quadrant? Update: Based on the ...
1
vote
1answer
270 views

Let p and q be distinct odd primes. Define $n=pq$ and$ \phi(n)=(p−1)(q−1)$

(a) Show that $p+q = n−\phi(n)+1$ and $p−q = \sqrt{(p+q)^2−4n}$. (b) Suppose you are given that $n = 675683$ and are told that $p−q = 2$. Explain how this information can help us factor $n$ ...
0
votes
1answer
49 views

Equality of corresponding variables

This question might be silly, but I was wondering for $x_1 y_1 + y_1 = x_2 y_2 + y_2$ , $y_1 = y_2$ is true. And likewise, $x_1 y_1 = x_2 y_2$ ?
1
vote
2answers
66 views

Solving for $y$ in $y= 14x + 1000 = y= 16x + 800$

Given $y= 14x + 1000 = y= 16x + 800$, solve for $y$. I think I have it: $x= 100$. Subbing that in would leave me with: $y = 14x + 1000$ $y = 14 \cdot 100 + 1000$ $y = 2400$ $y = 16x + 800$ ...
4
votes
1answer
38 views

Sequence of nonzero digits with sum dividing decimal representation

Is there an infinite sequence of nonzero digits $a_1,a_2,\ldots$ such that $$a_1+a_2+\ldots+a_n\mid\overline{a_1a_2\ldots a_n}$$ for all $n\geq 1$, where $\overline{a_1a_2\ldots a_n}$ denotes the ...
1
vote
4answers
86 views

Inequalities and absolute values

My book asks that if $$-5\leq x\leq 1$$ then find the boundaries of absolute value of $x$. Can you please help me in finding that?
0
votes
2answers
39 views

Change in square root?

I'm looking at an example problem where $\sqrt{1-(x^2/36)}$ is changed to $\sqrt{36-x^2}$ with no explanation. How does that work?
4
votes
2answers
55 views

Rational sequence with $a_{n+1}=2a_n^2-1$

Suppose we start with a rational number $a_0$, and define $a_{n+1}=2a_n^2-1$ for $n\geq 0$. For what $a_0$ will it be the case that $a_i=a_j$ for some $i\neq j$? We can start with something like ...
-1
votes
5answers
105 views

Solving a Quadratic Equation

$f(x) = 9x^2 - 48x + 14$ I need help in solving this equation. I cannot simply factorise it, so do I need to use the 'quadratic' formula to solve it?