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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9
votes
2answers
1k views

Why is the even root of a number always positive?

Let $n \in \mathbb N$ be a natural number and $a \in \mathbb R$ be a real number. The $n$-th root of the number $a$ is defined as follows: Case I: $n$ is an odd number. In this case the ...
2
votes
2answers
3k views

Convert any number to positive. How?

How to convert any number (negative or positive) into a positive number.. For example, my input can be 4545 or -4545, I need the out come to be positive value 4545.
3
votes
1answer
182 views

Subtracting rational functions

I'm trying to find out how to solve this: $$\frac{x-2}{x^2+2x} - \frac{x+2}{x^2-2x} - \frac{4x}{x^2-4}$$ The answer is $\displaystyle \frac{-4}{x-2}$ What is this called? And is there any video of ...
4
votes
3answers
213 views

How to show $ a,b,c \in \mathbb R , z \in \mathbb C , az^2 + bz + c = 0 \iff a\bar{z}^2 + b\bar{z} + c = 0$?

I am completely stumped by with this problem. If a, b and c are real numbers and z is a complex number of the form $x + yi$, prove that $$az^2 + bz + c = 0 \iff a(z^*)^2 + b(z^*) + c = 0$$ I ...
4
votes
4answers
439 views

Complex number to polar form

I need to take a raincheck with this problem. I want to make sure I haven't messed up some fundamental idea. Convert the complex number $$-\dfrac{1}{2} + \dfrac{\sqrt3}{2}i$$ to polar form. I ...
1
vote
3answers
589 views

Find opposite vertices of a rhombus, given the other 2

I am stuck with this problem. I posted an earlier problem with a square, where rotation with i of 90 degrees was possible. This one is a rhombus, how should I proceed? Given ABCD is a rhombus with ...
3
votes
4answers
229 views

Quadratic vertex form - negative $h$ means it's on the right side of the graph?

When putting a quadratic equation in vertex form, I am having difficulty understanding why $h$ is negative when the location of the parabola goes to the right... why is this? For example, if I ...
2
votes
4answers
445 views

Find remaining vertices of a square, given 2

I need a hint for this problem. Let the vertices of a square ABCD represent on the Argand diagram the complex numbers a,b,c, and d respectively. A,B,C,D are taken anti-clockwise in the order named. ...
7
votes
3answers
300 views

Confused by textbook solution to trig problem

The following is part of a question and the solution from my textbook: Question Given that $2 \sin{2\theta} = \cos{2\theta}$, show that $\tan{2\theta} = 0.5$ Solution $2 \sin{2\theta} = ...
1
vote
1answer
794 views

Distance of a point from a line passing through the intersection of lines

I'm given a point $P(1, 4)$ and its distance from a line passing through the intersection of the lines $x-2y+3=0$ and $x-y-5=0$ is 4 units. How do I find its equation? Here is how I tried to solve ...
18
votes
7answers
624 views

Solving $\sqrt{x+5} = x - 1$

I'm currently learning about radicals and simplifying them, and I came across this problem on the internet and tried to solve it: $$\sqrt{x+5} = x - 1$$ So I used this logic: $$ \begin{align} ...
1
vote
1answer
290 views

Is the logarithm function injective (one-to-one)?

Is the logarithm function injective (or, one-to-one)? In other words, does $\log_2(x) = \log_2(y) \implies x = y$? I.e., as $x$ and $y$ are in the same log base, can I just drop the logs? Thanks! ...
0
votes
3answers
147 views

Which of these variables affect y more?

I have two equations: 1: $\displaystyle y=\frac{0.0029}{\sqrt{k}}$ 2: $\displaystyle y=\frac{0.0060}{\sqrt{m}}$ Which variable has a greater effect on $y$? i.e., does doubling $m$ have a greater ...
1
vote
2answers
49 views

Indices confusion

Consider the equation $3^{y} = 9^{x}$ It follows that $3^{y} = 3^{2x}$ But $3^{2x} \equiv (3^{x})^{2} \equiv (3^{2})^{x}$ (I think? Since e.g. $(x^{2})^{3} \equiv x^{2 \cdot 3} \equiv (x^{3})^{2}$ ...
3
votes
2answers
233 views

Converting word problem into algebra

A microprocessor plant that manufactures computer chips costs $\$400,000$ per day to operate. It costs $\$30$ to produce one chip which sell for $\$190$ dollars. How many chips does the plant have to ...
2
votes
2answers
613 views

Did I sketch this polar curve correctly?

The equation is: $r^2=-4 \sin(2\theta)$ I first made a reference graph in cartesian coordinates using values $\displaystyle \frac{\pi}{4}$, $\displaystyle \frac{\pi}{2}$, $\displaystyle \frac{3 ...
10
votes
5answers
510 views

imaginary numbers - how can I understand them - especially as they occur in 'roots' of polynomials?

In another question here, about roots of equations being imaginary, the accepted answer said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I ...
5
votes
1answer
161 views

Define term using Stirling Numbers

I am trying to solve the following exercise - quite unsuccessful yet. Let a(m,n) be defined as $$ \sum\limits_{n=0}^m a(m,n) \prod\limits_{i=1}^n (x+i-1) = x^m $$ Express a(m,n) using S(m,n) ...
1
vote
1answer
77 views

Of two variables, which affects 'y' more?

I have two equations: (1) $\displaystyle y = \frac {0.0060}{k}$ (2) $\displaystyle y= \frac{0.00016}{m}$ In the first equation, $k$ is held constant and in the second equation, $m$ is held ...
2
votes
2answers
94 views

Finding, writing down and visualizing images of functions

I apologize in advance if this is a duplicate (I would expect to see something like this on here), but I haven't found anything that matches my question. I have this general difficulty (perhaps, to ...
0
votes
1answer
107 views

Why is 9 involved in this problem instead of 3?

So, lets say it costs $1.25 per square yard to paint a wall. The wall is 18x25 feet. How much will it cost to paint the wall. First, I don't understand what is better. Converting 18x25 feet into ...
2
votes
2answers
571 views

How do I average out to a particular number if I have to add another number to get that average?

So, lets say, that you have the following numbers (4, 15, 3, 22) and you have to add the fifth number (which I would call X) and they must average out to say 20. What do you have to do? I figured ...
2
votes
2answers
126 views

Percentage question

Certain alloy contains 20% of copper and 5% of tin. How much of copper and how much tin must be added to 100kg of this alloy to get another with 30% of copper and 10% of tin? I can't figure how to ...
6
votes
4answers
275 views

Trig equation help please

I am trying to solve $\sqrt{3}\tan\theta=2\sin\theta$ on the interval $[-\pi,\pi]$. $$\sqrt{3}\tan\theta=2\sin\theta \Rightarrow \sqrt{3}=\frac{2\sin\theta}{\tan\theta}$$ $$\Rightarrow ...
3
votes
3answers
612 views

How do get combination sequence formula?

What would be a closed-form formula that would determine the ith value of the sequence 1, 3, 11, 43, 171... where each value is one minus the product of the previous value and 4? Thanks!
5
votes
3answers
103 views

For what $a$ and $b$ is $9x^4-12x^3+28x^2+ax+b$ a perfect square?

If $9x^4-12x^3+28x^2+ax+b$ is a perfect square, find the value of $a$ and $b$. This is one of my past year examination's questions, some help on it? (The answer for this problem is $a=-16$, $b=16$.)
0
votes
1answer
581 views

Solving an equation with the same variable twice

This may be kind of obvious, but our finance lecturer just assumes we know how to do stuff like this and the slides give no explanation. Jumping from question to answer How do I get $r$ from this? ...
6
votes
2answers
287 views

Really easy algebra problem

Can somebody show me how to go about solving the following (really easy) equation for $\alpha$ please? $$\displaystyle 0 = \sum_{n=1}^{N}\left( \frac{y_n}{\alpha} + \frac{1-y_n}{1-\alpha}\right)$$
11
votes
8answers
491 views

How to factor quadratic $ax^2+bx+c$?

How do I shorten this? How do I have to think? $$ x^2 + x - 2$$ The answer is $$(x+2)(x-1)$$ I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link to ...
1
vote
2answers
175 views

Multiplying Binomial Terms

$(x+2)(x-1$) Totally forgot how to do it. Isn't this the rule: $(a+b)(c+d)$ $a*c +a*d +b*c +b*d$ The answer I'm getting is $x^2-x+2x-2$ which is wrong. Please help me out, thanks!
2
votes
1answer
117 views

Bounding a Complex Polynomial

Given the complex polynomial $P(z) = z^2 + a_1z + a_0$ and the constraint that $|z| > 1$, I'm trying to show that $|P(z)| \geq |z|^2 - |a_1||z| - |a_0|$. The obvious thing to do here of course is ...
-2
votes
3answers
281 views

How to solve $z^3\bar{z}+4i=2z(z+i\bar{z})$ in $\mathbb{C}$

I need to solve the complex equation $z^3\bar{z}+4i=2z(z+i\bar{z})$
1
vote
4answers
175 views

Calculating $a^p+b^p$ given $a^2+b^2=c$

If one has $a^2+b^2=c$ where $a$,$b$ and $c$ are real numbers, is there any way to calculate $a^p+b^p$ where $p$ may be any real number? If there is, would you please explain with an example with ...
3
votes
2answers
140 views

Calculate max/min of $x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$

What is a good way to calculate max/min of $$x_1 x_2+y_1 y_2+z_1 z_2+w_1 w_2$$ where $x_1+y_1+z_1+w_1=a$ and $x_2+y_2+z_2+w_2=b$ and $x, y, z, w, a, b \in \mathbb{N} \cup \{0 \}$, and please explain ...
5
votes
3answers
3k views

How to find roots of $X^5 - 1$?

How to find roots of $X^5 - 1$? (Or any polynomial of that form where $X$ has an odd power.)
1
vote
2answers
104 views

Work out the percentage of a single unit in a scale

1st question on this forum, and I know it will be an easy-peasy one for someone out there!! I'm not even sure if the title of the question is accurate for the question I am asking.....I'm also not ...
4
votes
1answer
338 views

Two proofs involving Harmonic Mean

If H is the harmonic mean between $a$ and $b$,then show that $$\frac{1}{H-a}+\frac{1}{H-b} = \frac{1}{a} + \frac{1}{b}$$ and $$\frac{H+a}{H-a}+\frac{H+b}{H-b} = 2$$ I substituted $\displaystyle H = ...
2
votes
3answers
690 views

Simple gradient/line intersect question

Very, very basic question here: Given an x,y coordinate and a gradient (but no equation), how can I find the x and y axis intercepts? (assuming the line is linear)
2
votes
4answers
1k views

Calculating acceleration based on a starting and ending distance

I have been trying to figure this problem out and just can't. The answer is supposed to be (A): 1.0 m/s/s. Can anyone tell me how they got that answer? I thought this problem was based on the ...
3
votes
1answer
184 views

Devise System of Equations to Solve Age Problem

Hi this may be simple silly problem but it is bugging me as I am not able to devise a system of equations to solve it. My husband's age," remarked a lady the other day, "is represented by the ...
9
votes
3answers
7k views

Equation of angle bisector, given the equations of two lines in 2D

I have two lines in 2D expressed with general equation (or implicit equation): First line: $a_1x+b_1y=c_1 \qquad(1)$ Second line: $a_2x+b_2y=c_2 \qquad(2)$ If the two lines are intersecting I will ...
14
votes
4answers
492 views

How to prove $\sum\limits_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$?

Other than the general inductive method,how could we show that $$\sum_{r=0}^n \frac{(-1)^r}{r+1}\binom{n}{r} = \frac1{n+1}$$ Apart from induction,I tried with Wolfram Alpha to check the validity,but ...
-2
votes
2answers
136 views

Please solve for X

I messed up the equation last time I asked this - Can someone please solve this function for X? $Y = \displaystyle 0.5\:a\:\text{erf}\left(\frac{x-b}{c\sqrt{2}}+.5\right)+d$ When I solve for Y with ...
0
votes
1answer
133 views

Mathematical proofing II [closed]

The roots of a function $f(x)$ are the solution to the equation $f(x)=0$. Utilize factoring and quadratic formula to show that $f(x) = x^2 - 2x + 5$ has no real roots. Your help will be appreciated ...
0
votes
2answers
48 views

X and Y to smooth destination

I got 4 Variables: X Y TargetX TargetY Basically X and Y are my start Point and I want to get to The Target. I increase ...
1
vote
4answers
7k views

Factoring Methods/Tricks

One of the things I've struggled with most in algebra/calculus is all the "factoring tricks". When I take time away from doing math I inevitably forget most if not all of them. The old proverb "use it ...
1
vote
1answer
184 views

What are logarithms, and what do they do? [duplicate]

Possible Duplicate: Intuitive use of logarithms My math teacher "taught" us about logarithms today, but he didn't give any useful information. He just that one is supposed to "add" them to ...
2
votes
3answers
264 views

How long do I have to own my Hybrid Prius in order to see the saving?

I purchased the Prius for 20k and goes 45 mpg. The other car costs 15k and goes 25mpg. What is the formula to calculate how long do I need to keep my car in order to see the saving. Number of miles ...
1
vote
1answer
598 views

Why is the compound interest formula exponential?

I was trying to answer a programming challenge about interest, and my first thought was to model the situation using a single recurrence: principal = (principal - payment) * interest_rate; The ...
2
votes
1answer
487 views

Describing an equation

Which of the following describe this formula correctly: $S = 2\pi rh$ 1.S is jointly proportional to radius and height 2.S is directly proportional to radius and height or 3.S varies directly with ...