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
17 views

Graphing a parabola

In the video, he shows $5x^2-20x+15$ is divided by $5$ across every term and becomes $x^2-4x+3$. Then, he proceeds to factorize the equation to $(x-3)(x-1)$ to get the roots $(1,0)$ and $(3,0)$. ...
0
votes
1answer
15 views

Let $z_1$ and $z_2$ be the $nth$ roots of unity which subtend a right angle at origin, then prove that n must be of the form $4k$

Problem : Let $z_1$ and $z_2$ be the $nth$ roots of unity which subtend a right angle at origin, then prove that n must be of the form $4k$ Solution : Here $arg \frac{z_1}{z_2}=\frac{\pi}{2}$ ...
-3
votes
1answer
29 views

Find the number of bricks needed to cover a given space [closed]

I have a space the size of $\,72$" $\times 36$"$,$ and bricks of the size of $12$" $\times 12$". How many bricks would be needed to cover this space?
0
votes
1answer
14 views

Time And Distance (Train Journey)

The average speed of a train in the onward journey is 25% more than that in the return journey. The train halts for one hour on reaching the destination. The total time taken for the complete to and ...
0
votes
1answer
35 views

Time And Distance (Gunshots and Train) [closed]

Two guns were fired from the same place at an interval of 10 minutes and 30 seconds, but a person in the train approaching the place hear the second shot 10 minutes after the first. Speed of sound is ...
3
votes
8answers
214 views

What is the value of $i^0$?

I have to solve the following question - $$\sum_{n=0}^{1000} i^n$$ where $i = \sqrt{-1}$ To be able to solve the problem, I need to know the value of $i^0$. What is the value of $i^0$? Is it 0 or ...
1
vote
1answer
24 views

Derivative of a polar coordinate equation

I was trying to plot the polar curve: $r=\cos(2n\theta)$ ($0\leq\theta\leq 2\pi$) and tried differentiating with respect to $\theta$ to get some information about where the petals would be. My ...
0
votes
0answers
18 views

Reduce expresion

How can I reduce this the following expresion $$\sum_{k = 1}^m \frac{x_1 \sin \left( \frac{2k\pi}{m} \right) - x_2 \cos \left( \frac{2k\pi}{m} \right)}{x_1 \cos \left( \frac{2k\pi}{m} \right) + x_2 ...
-2
votes
4answers
61 views

Find $x- \frac 1x$ when $x + \frac 1x = 3$

When $$x + \frac 1x = 3,$$ Evaluate the exact value of $$x - \frac 1x .$$ PS. I know the answer is $\pm\sqrt 5$.
0
votes
1answer
33 views

Expression of Vectors

(a) Show that any two-dimensional vector can be expressed in the form $s \begin{pmatrix} 3 \\ -1 \end{pmatrix} + t \begin{pmatrix} 2 \\ 7 \end{pmatrix},$ where $s$ and $t$ are real numbers. (b) Let ...
0
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0answers
14 views

Converting a polar equation in all variables to find properties of the corresponding linear equation.

Given the equation $\sin R = nx^q$, find the slope, $m$ and $y$-intercept, $c$, that corresponds to the linear form ($y=mx+b$) of the same equation. I understand this is a trivial question I am ...
2
votes
2answers
31 views

Algebraic Manipulation in the Proof of Heron's formula

A textbook I'm reading gives a proof of Heron's formula, but has lost me in one of its steps. My mathematical foundations are a bit shaky, so I was hoping someone could explain what was done. The jump ...
7
votes
2answers
71 views

Solve $(2+\sqrt{3})^{x/2}+(2-\sqrt{3})^{x/2}=2^x$.

How to solve $(2+\sqrt{3})^{x/2}+(2-\sqrt{3})^{x/2}=2^x$ for $x$?
3
votes
1answer
40 views

Determine the amplitude and phase shift of $f(x) = \sqrt{3} \cos2x-\sin2x$

Question: Determine the amplitude and phase shift of $f(x) = \sqrt{3} \cos2x-\sin2x$ Attempted solution: The amplitude can be calculated by: $$A = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{4} = 2$$ ...
-1
votes
2answers
55 views

Real roots of a non linear equation

Determine the number of real roots of the equation- $$x^2+10x+(\sin(x))^2=\cos(x)$$ How to solve such kind of questions? Please suggest some reference also.
6
votes
6answers
498 views

What is the purpose of the compound angle identity in trigonometry?

This may be a silly question, but one that I am confused about nonetheless. With regards to the compound trig identities such as $\cos(A+B)=\cos A\cos B - \sin A\sin B$ etc., I'd like to know why ...
2
votes
5answers
116 views

Solving $ \sqrt{x - 4} + \sqrt{x - 7} = 1 $.

I have the equation $ \sqrt{x - 4} + \sqrt{x - 7} = 1 $. I tried to square both sides, but then I got a more difficult equation: $$ 2 x - 11 + 2 \sqrt{x^{2} - 11 x - 28} = 1. $$ Can someone tell me ...
1
vote
2answers
40 views

How to prove this logarithm equation?

Given : $$\log_{12}18 = a \text{ and }\log_{24}54=b$$ prove that: $$ab + 5(a-b) = 1$$ My attempt: I couldn't solve it in any way, as base were not common. I could solve it if base of second ...
1
vote
1answer
22 views

Time, Speed and Distance

A walks around a circular field at the rate of one round per hour while B runs around it at the rate of six rounds per hour. They start in the same direction from the same point at 7.30 a.m. They ...
5
votes
2answers
45 views

A trigonometric product

I have to prove: $$\prod_{i=1}^6 \left(2\cos\left(\frac{2^{i}\pi}{13}\right)-1\right)=1$$ I really have no idea about starting with this one. With the help of Wolfram Alpha, I noticed that: ...
3
votes
3answers
47 views

Having a difficult time figuring out if linear equations is true or false.

I'm having a very difficult time figuring out if these linear equations are true or false: $$ x + 2y > 6\\ x - y < 3 $$ How do you identify if these equations are true or false? At first I ...
1
vote
3answers
50 views

What happens with a sign?

I have something simple to ask, but it often confuses me. What happens with number-sign when it crosses over the equal sign or less-than and greater-than sign. I have an example. $$1-3x\geq0$$ ...
5
votes
3answers
53 views

Show $x^n \geq x_1^n+x_2^n+\ldots+x_k^n \Bigg\vert x_1+x_2+\ldots+x_k = x, x \geq 0, n \geq 1, k\in\mathbb{Z}$

Hello StackExchange Community, This is my first post on the forum. Please forgive me for any errors with formatting and my expressions. I am working on the following proof: Show $$x^n \geq ...
1
vote
1answer
24 views

Amplitude and phase shift of a periodic function

I am stuck on this particular question... I've attached an image of the problem since I am not well versed in typing math equations. I'm assuming that I will need to use the sum and difference ...
0
votes
2answers
74 views

Solve $2\sqrt{1-\frac{2}{x}}+\sqrt{x-\frac{4}{x}}=x$

Solve this equation $$2\sqrt{1-\tfrac{2}{x}}+\sqrt{x-\tfrac{4}{x}}=x$$ I tried to solve on my computer, but then it has a root that is not too nice. Can you help me?
0
votes
0answers
31 views

Regarding penalty of choosing points..

In my precalc course, I am required to complete a rather lengthy project regarding least squares. I have gotten stuck on a particular question which asked "Refer to a scatter plot with the ...
0
votes
1answer
20 views

How would I solve for a rate that compounds m times per annum?

Please excuse me, this is my first time using the site and I have absolutely no idea what I'm doing with the notation. Anyways, I am attempting to prove that: $$R_m = ...
2
votes
3answers
33 views

partial fraction decomposition help now

very quick way to solve 2x/x^2-4 as a partial fraction. I have tried the long way and it took over 30 minutes, I got it right but is it easier another way?
2
votes
2answers
48 views

Finding conditions to make roots of a quadratic less than one in magnitude

I'm doing a problem that asks for you to find the conditions that make $y$ defined: $$y=x^2-bx+c$$ have real roots with magnitude less than one. Now the condition for the roots being real seems to ...
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5answers
61 views

Quadratic solutions puzzle

The equation $x^2+ax+b=0$, where $a\neq b$, has solutions $x=a$ and $x=b$. How many such equations are there? I'm getting $1$ equation as I can only find $a=b=0$ as an equation, which is not allowed. ...
2
votes
1answer
84 views

Breaking up a radical

A common misconception in math is breaking up a radical, i.e., suppose we have $$a^n + b^n = c^n \qquad \qquad (1)$$ and we take the square root of both sides, but incorrectly for the LHS: ...
3
votes
3answers
49 views

Roots of unity are distincts

For every $n\in\Bbb N$ and $$z_{k}:= \cos(2\pi k /n)+i\sin(2\pi k /n), \qquad k = 0,\ldots,n-1$$ we have $z_k^n=1$. How to show, in a simple way, that $z_k\neq z_l$ for every $k\neq l$? By ...
-1
votes
0answers
54 views

Even or Odd periodic functions

Are the following functions, which are assumed to be periodic and of period $2\pi$, even or odd or neither? $f(x)= x^2$ ($0 < x < 2\pi$) $f(x)= |\sin 5x| $ ($-\pi < x < \pi$) I ...
2
votes
2answers
52 views
+50

Integer reciprocals in arithmetic progression

Let $m_1,m_2,\ldots,m_k$ be $k$ positive integers such that their reciprocals are in AP. Show that $k<m_1+2$. Also find such a sequence. Whatever way I tried, whichever formula I used, I ...
-2
votes
4answers
66 views

Working out my exam grade [closed]

Okay, this may be a simple question but I have an exam that is 70% coursework and I got an A, Is it possible to work out how much I will need to get in my 30% exam to achieve an A overall
1
vote
1answer
69 views

$f(x)=\sum_{i=0}^{\infty} (x^{2^n})/(1-x^{2^{n+1}})$. Find $f(99)$.

$f(x)=\sum_{i=0}^{\infty} (x^{2^n})/(1-x^{2^{n+1}})$. Find $f(99)$. ATTEMPT: The following series can be re-written as $f(x)=\sum_{i=0}^\infty \left(\frac{1}{1-x^{2^n}}\right) \cdot \left( ...
1
vote
3answers
72 views

Prove/disprove an inequality $\sqrt{1 + x} < 1 + \frac{x}{2} - \frac{x^2}{8}$

Is $\sqrt{1 + x} < 1 + \frac{x}{2} - \frac{x^2}{8}$ TRUE in $(0, \frac{π}{2})$ ? I proceed by taking two functions $f(x) = \sqrt{1 + x}-1 $ and $g(x) = \frac{x}{2} - \frac{x^2}{8} $. Then $f(0) ...
1
vote
3answers
48 views

Linearizing an equation containing both $x$ and $\ln x$

The equation of interest is of the form: $$ k_1 \ln(y/x) = k_2 x $$ And I am wondering how can one linearize this equation for $x.$ Splitting the $\ln$ function would give something along: $$ k_1 \ln ...
3
votes
2answers
25 views

how can I write this statement in equation form?

If $x$ men take $5$ days to reap a fields, how long will one men take? I know its answer will be $5x$ but I don't know how to write this statement in equation form. Thank you in advance!
2
votes
2answers
55 views

Find numbers $a, b, c$ given that $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$

Let $a+b+c=12$, $a^2+b^2+c^2=50$, and $a^3+b^3+c^3=168$. Find $a,b,c$ Suppose $a, b, c$ are roots of $P(x)$. $$P(x) = k(x - a)(x - b)(x - c)$$ But then I get $(k = 1)$ $$P(x) = x^3 - 12x^2 + ...
1
vote
2answers
37 views

Roots are the reciprocal of $f(x)$

I don't understand if $f(x)$ has roots, $r_1, r_2$ for example and $g(x)$ has roots $\frac{1}{r_1}, \frac{1}{r_2}$ Then how is $g(x) = x^2f(\frac{1}{x})$ What does $$f(\frac{1}{x})$$ have to do ...
2
votes
4answers
71 views

Solve: $f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x).f(\frac{1}{y})$

Here i have one functional equation: If $$f(x+\frac{1}{y}) + f(x-\frac{1}{y}) = 2f(x)\cdot f(\tfrac{1}{y})\text{ for all x},y\in\mathbb{R}-{0}$$ and $f(0) = \frac{1}{2}$ , then find the value of ...
2
votes
1answer
47 views

How to know what kinds of substitution can we do in math?

I have seen in many contexts that somebody out of the blue decides to put $x=y^2$, or $x=t/2$. So how do I know what kind of sustitution I'm allowed to do? Is there any necessary conditions or we ...
2
votes
3answers
33 views

Are the sum and/or product of two increasing functions also increasing?

Question: Let $f(x)$ and $g(x)$ be two increasing functions. a) Show that their sum is also increasing. b) Investigate the corresponding claim for the product of two increasing functions. ...
-6
votes
3answers
61 views

What is the solution of $a^2=b^2$? [closed]

How to solve $a^2=b^2$? Should I consider if the number is negative or positive?
0
votes
1answer
25 views

Quick formula rearranging

I'm having problems rearranging this formula to solve for c, could someone lend a hand please. It's a physics formula for projectile motion. ...
6
votes
1answer
75 views

Prove this Complicated Inequality

Let $a$, $b$, $c$ be positive real numbers such that $a^2 + b^2 + c^2 + (a + b + c)^2 \le 4$. Prove that $$\frac{ab + 1}{(a + b)^2} + \frac{bc + 1}{(b + c)^2} + \frac{ca + 1}{(c + a)^2} \ge 3.$$ ...
2
votes
1answer
42 views

Factorising polynomials over $\mathbb{Z}_2$

Is there some fast way to determine whether a polynomial divides another in $\mathbb{Z}_2$? Is there some fast way to factor polynomials in $\mathbb{Z}_2$ into irreducible polynomials? Is there a ...
26
votes
8answers
832 views

Why is $1/i$ equal to $-i$?

When I entered the value $$\frac{1}{i}$$ in my calculator, I received the answer as $-i$ whereas I was expecting the answer as $i^{-1}$. Even google calculator shows the same answer (Click here to ...
0
votes
1answer
26 views

Sum of products with K elements with different indexes

Is there a fast way to find a $SUM(N,K)$ where we define $SUM(N,K)$ to be $$SUM(N,K)=\sum a_{i1}a_{i2} \cdots a_{ik},$$ where $a_{ij}$ are $K$ numbers chosen from $N$ numbers. For example ...