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
votes
3answers
43 views

The absolute maximum of a function

Let $f(x)=\dfrac{\sqrt{4+32x^{2}+x^{4}}-\sqrt{4+x^{4}}}{x}$, where $x\in \mathbb{R}$ and $x\neq 0$. Suppose that $f(x_{0})=M$ is the absolute maximum of $f$. Find $(x_{0},M)$. I have no good ideas. ...
12
votes
4answers
576 views

Simplifying $\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt {5 +\cdots}}}}$

How to simplify the expression: $$\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt{\cdots}}}}.$$ If I could at least know what kind of reference there is that would explain these type of expressions that would be ...
0
votes
2answers
99 views

Changing the subject of an equation

How can we change the subject of the equation from $x \to y$? For $$x^{3/2}=y^{3/2}+ay^{-1/2}$$ where $a$ is some constant?
1
vote
1answer
103 views

How to solve the equation $\sqrt{x^{2}-x+2}+\sqrt{2x^{2}+2x+3}=\sqrt{2x^{2}-1}+\sqrt{x^{2}-3x-2}$

Solve in real numbers the equation $$\sqrt{x^{2}-x+2}+\sqrt{2x^{2}+2x+3}=\sqrt{2x^{2}-1}+\sqrt{x^{2}-3x-2}.$$
6
votes
8answers
1k views

Solving $\sin \theta + \cos \theta=1$ in the interval $0^\circ\leq \theta\leq 360^\circ$

Solve in the interval $0^\circ\leq \theta\leq 360^\circ$ the equation $\sin \theta + \cos \theta=1$. I've got the two angles in the interval to be $0^\circ$ and $90^\circ$, it's not an answer I'm ...
1
vote
1answer
64 views

Help evaluating $\lim_{\left|x\right| \to \infty}y$, given $\frac{y^2}{x^2}=\frac{b^2}{a^2}-\frac{b^2}{x^2}$

I'm trying to understand this step in a derivation of the standard equation of a hyperbola. We have constants $a$ and $b$, and variables $x$ and $y$. We've gotten to a point where we have ...
2
votes
1answer
75 views

How to solve $[x+1]=0$ for $x\in (0,1)$

Let $[x]$ be the integer part of the real $x$. How can we solve the algebraic equation $[x+1]=0$ for $x\in (0,1)$?
3
votes
5answers
2k views

Prove that $\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$

$$\cos (A + B)\cos (A - B) = {\cos ^2}A - {\sin ^2}B$$ I have attempted this question by expanding the left side using the cosine sum and difference formulas and then multiplying, and then ...
9
votes
3answers
487 views

Being ready to study calculus

Some background: I have a degree in computer science, but the math was limited and this was 10 years ago. High school was way before that. A year ago I relearnt algebra (factoring, solving linear ...
3
votes
4answers
150 views

Where have I made my mistake in my integrating $\int x \sqrt{2x - 1} dx$?

$$\int x \sqrt{2x - 1} \,dx$$ Let $u = 2x - 1$ $$\int x \sqrt{u}\, dx$$ $$\frac{du}{dx} = 2 \implies \frac{1}{2}du = dx.$$ So the integral is written as $$\int \frac{1}{2} u^{\frac{1}{2}} \, du$$ ...
1
vote
1answer
60 views

How to determine if an empirical probability is statistically different to 50/50?

I have $431$ observations of nursing care and am interested to know if touching a patient influenced their handwashing compliance. I have a column for Handwash (Yes/No) and one for Patient contact ...
-1
votes
5answers
12k views

$\cos(\arcsin(x)) = \sqrt{1 - x^2}$. How?

How does that bit work? How is $$\cos(\arcsin(x)) = \sin(\arccos(x)) = \sqrt{1 - x^2}$$
1
vote
2answers
121 views

Proving that $\sin(a)\cos(b)$ and $\cos(a)\sin(b)$ identities are identical using $\sin(-x)=-\sin(x)$

This website states the two trig identities below are identical: [\begin{array}{l} \sin (a)\cos(b) = \frac{1}{2}\left[ {\sin (a + b) + \sin (a - b)} \right] \Rightarrow 1\\ \cos (a)\sin (b) = ...
3
votes
0answers
135 views

Solve this equation $ (x^3+100)^2 = (x^4-100)^3$ [duplicate]

Solve this equation $(x^3+100)^2 = (x^4-100)^3$
3
votes
1answer
94 views

Integer triples $(x,y,z)$ satisfying $xyz+4(x+y+z)=2(xy+yz+zx)+7$

How can we find integer triples $(x,y,z)$ satisfying $xyz+4(x+y+z)=2(xy+yz+zx)+7$?
1
vote
3answers
53 views

How to compute this geometric progression?

I am wondering how to compute this variant of geometric progression: $$ \sum_{i,j \in \{1, \dots, n\}, i \neq j} c^{i+j}? $$ Any help is appreciated!
2
votes
1answer
133 views

License plate consisting of 4 letters and 4 numbers

While doing homework today, the following question popped into my head: Can you easily calculate the amount of unique license plates consisting of 4 letters and 4 numbers in any order? It doesn't ...
0
votes
3answers
67 views

Solving the algebraic equation

I am trying to solve this: $$x-40={-400\over x}$$ The answer must be $x=20$ Please give step by step explanation.
0
votes
2answers
48 views

Trouble with simplication in algebra

I am having trouble with $$((x-1)+x^2)((x-1)-x^2)=$$ Can I just ignore the inside bracket? And thus the question becomes like this? $$(x-1+x^2)(x-1-x^2)=$$ Che second equation should not have ...
3
votes
3answers
85 views

Why infinity has minus sign in notation $(-\infty,b)$

Why is the convention $(-\infty,b)$ used to represent the interval set of all the real numbers $x$ which are less than $b$? We could write infinity without the minus sign. Why is the minus sign ...
3
votes
2answers
197 views

Why $x^{(1/2)2} \neq x^{2(1/2)} $?

I know, probably is a newb. question, but i can't get this $x^{(1/2)2} \neq x^{2(1/2)} $ $ x\in\mathbb R^+$. I know $x^{(1/2)2}=(\pm \sqrt{x})^2=+x $ and $x^{2(1/2)}=\pm x $ because ...
4
votes
2answers
97 views

If $(\pm x)^2 = X$ then $\sqrt{X} = x$?

I've been helping my siblings with their GCSE and A Level maths and I've come across a question where they have just taken the positive square root. It's a pure maths question and there's no (obvious) ...
1
vote
4answers
491 views

The least possible value

How to find the least possible value for :$$(x-1)^2+(x-2)^2+(x-3)^2+(x-4)^2+(x-5)^2$$ For every real $x$
3
votes
4answers
143 views

Can $\frac{a}{b} + \frac{b}{a}$ ever be bounded from above, if $a, b \in \mathbb{R}$?

Good morning everyone! By the arithmetic-geometric mean inequality, we all know that a suitable lower bound for the quantity $$\frac{a}{b} + \frac{b}{a}$$ is $2$. Now my question is: Will this ...
0
votes
2answers
75 views

Simplifying $\frac{4}{2x-7}-\frac{3}{(2x-7)^2}$

A homework question asks me to "perform the addition or subtraction and simplify" $$ \begin{gather} \frac{4}{2x-7}-\frac{3}{(2x-7)^2}=4(2x-7)-3=8x-28-3=8x-31 \\ 8x=31 \\ x=\frac{31}{8} \end{gather} ...
7
votes
2answers
93 views

factorisation of algebra

homework question that asks to perform the multiplication and division and simplification. $$\frac{x^2+7x+10}{x^2+5x+4} \times \frac{x^2+3x+2}{x^2+4x+4} =$$ $$\frac{(x+5)(x+2)}{(x+4)(x+1)} \times ...
10
votes
5answers
936 views

Factor $(a^2+2a)^2-2(a^2+2a)-3$ completely

I have this question that asks to factor this expression completely: $$(a^2+2a)^2-2(a^2+2a)-3$$ My working out: $$a^4+4a^3+4a^2-2a^2-4a-3$$ $$=a^4+4a^3+2a^2-4a-3$$ $$=a^2(a^2+4a-2)-4a-3$$ I am ...
0
votes
2answers
65 views

Finding the values of x and y from given condition

If $x^y = y^x$ and $x-y=2$, find the value of $x$ and $y$. Please give me the necessary steps too!
0
votes
2answers
310 views

How to count angle between vector and horizontally oriented vector?

I need to calculate in my Java application an angle between my line and horizontal line that has the same beginning. I have a line described by its equation: $$f(x) = ax + b.$$ I would like to know ...
1
vote
2answers
40 views

Flipping variables in function

I'm in a bit of dilemma. I have the following formula $$ F_{b} = \frac{R_{b} - R_{0}}{R_{b} + R_{0}} $$ Variable $ F_{b} $ and $ R_{0} $ are known to me how can i pull $ R_{b} $ out so i can ...
1
vote
1answer
47 views

Linear decrease

I have a problem of retained sales from the previous year. For instance, x client sell 200 cases of product the first month, 400 the second month, 300 the third and so on. The next year's sales are ...
1
vote
1answer
55 views

How best to display a table of probabilities to a non-mathematical audience?

How would you best display this table of probabilities in an explanatory document? Basically they are the probibalbities of touching one of euipment, patient, hygiene products, etc given a type of ...
0
votes
2answers
179 views

Matrices Problem with 3 Unknown Variables J, K and M

Given: $$ \left[\matrix{1&3 \\-2&4 }\right]+ \left[\matrix{11&5 \\-6&12 }\right]=K\left(\left[\matrix{3&2 \\J&M }\right]\right) $$ Find the value of $J+K+M.$ the answer is $6$ ...
3
votes
1answer
83 views

Why does (x-y/2)&y plot a sierpinski triangle?

The function (x-y/2)&y = 0 can be used to plot a sierpinski triangle. For example: Why does this happen?
3
votes
1answer
326 views

Solution of a nonlinear system of equations

How to solve this system of equations of second degree in set of real numbers $$15(x+y)=(15+x+y)x$$ $$(15+x+y)x=12\sqrt{x^2-6^2}+9\sqrt{y^2-(9/2)^2}+108 $$
1
vote
3answers
170 views

Why does tan(t) touch the unit circle at (1,0)?

I can't get my head around this, any help would be very much appreciated. Thanks EDIT: t is an angle, where 0 < t < 90, angle t is in degrees EDIT: Added a picture I lifted from google
0
votes
1answer
145 views

Area of intersection of two general overlapping circles

My algebra is letting me down here, I can't figure out how to arrange this equation - anyone prepared to give me a hand? The area of the intersection of two circles can be defined as $$A = r^2 ...
2
votes
2answers
485 views

Strong Induction: Prove provided recurrence relation $a_n$ is odd.

I'm not sure if we're allowed to post pictures but I thought it would be easier to read and I didn't see anything in the rules about it. It's question 1. Section 5.4 This question: Here is the ...
5
votes
3answers
140 views

Given $a+b+c=0$, simplify the following.

I am here again to ask a question about an exercise I saw around but i'm having a lot of trouble with. I know the answer is 3abc, but as in many of my questions, I am interested in the why and how. ...
3
votes
2answers
36 views

Does this equality hold?

Is the following proposition true for real numbers? $$\frac{\sqrt{2|x|-x^2}-0}{x}=\begin{cases}\sqrt{\frac{2}{x}-1}&,\;\;\;x>0\\{}\\\sqrt{-\frac{2}{x}-1}&,\;\;\;x<0\end{cases}$$ I ...
1
vote
5answers
3k views

In the context of the Unit Circle why is tan$(\theta)$ defined as $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$?

I understand why the circular functions $\sin(\theta)=y$ and $\cos(\theta)=x$, but why does $\tan(\theta)=\frac{\sin (\theta)}{\cos (\theta)}=\frac{y}{x}$? Is there any particular reason why $\tan ...
1
vote
5answers
102 views

Every polynomial of degree $\le m$ can be written in a form using its $m+1$ values

I'm not sure if the title is comprehensible. What I mean is this: I've found here the following corollary: Let $x_1, ..., x_{m+1}$ be $m+1$ different points in $\mathbb{R}$, and for $i = ...
3
votes
3answers
177 views

Proving that there are relatively prime elements (Posa problem) [duplicate]

How to prove that for each $n+1$ an element from the set $\{1,2,3,4,\ldots,2n-1,2n\}$ there are two relatively prime elements between them
0
votes
2answers
108 views

Aproximate calculation in decimals

I am trying to refresh on precision of calculations. If we have the decimal fractions: $.234673$, $.322135$, $.114342$, $.563217$ each known to be correct to six figures why are each of the decimals ...
0
votes
1answer
123 views

Relationship between three variables

I have three variables $r$, $p$, and $ w$. The $w$ is weight, $r$ is distance, and $p$ is the number of people. Morover, I have $$r = aw^b,~~p = ae^{bw}$$ where $a$ and $b$ are constants. Given a ...
1
vote
0answers
87 views

What are the constants in the relationship between point density and median point distance?

Given $\rho$ particles uniformly distributed on a plane within a unit square ($\rho > 1$), each particle has another particle that is closest to it; the median of those nearest distances is called ...
9
votes
2answers
297 views

Algebraic equation problem - finding $x$

$$(x^2 +100)^2 =(x^3 -100)^3$$ How to solve it?
1
vote
3answers
123 views

Diagonal of a rectangle

I need help solving this problem: The diagonal of a rectangle is $18$ cm longer than the shorter leg. If the area is $168 \ \text{cm}^2$, find the dimensions of the rectangle.
3
votes
3answers
119 views

Evaluating $\sum^{7}_{k=1}\frac{1}{\sqrt[3]{k^2}+\sqrt[3]{k(k+1)}+\sqrt[3]{(k+1)^2}}$

How do I evaluate this? $$\sum^{7}_{k=1}\frac{1}{\sqrt[3]{k^2}+\sqrt[3]{k(k+1)}+\sqrt[3]{(k+1)^2}}$$ I know I'm suppose to use substitutions but didn't work out so far. Help anyone?
2
votes
2answers
56 views

Looking at the numerator first or denominator?

If I wanted to work out the range of values for something like $$\frac{xy}{x + y},$$ then what do I take into consideration first? Clearly, if $x = y = 0$, then the denominator is $0$ which is "not ...