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
1answer
69 views

Is this (or when) does this equality hold?

Let $a,b,c,d \in \mathbb{R}$ and $x,y$ are variables which are also real numbers $$|ax + by|^2 + |cx + dy|^2 + 2|ax + by||cx + dy| = (ax + by)^2 + (cx + dy)^2 + 2(ax + by)(cx + dy)$$ Is this always ...
2
votes
0answers
65 views

Clock question?

On Saturday, Jimmy started painting his toy helicopter between 9:00 a.m. and 10:00 a.m. When he finished between 10:00 a.m. and 11:00 a.m. on the same morning, the hour hand was exactly where the ...
2
votes
2answers
651 views

Pre-Calculus Vector Problem.

In this question vector i represents a vector due east and vector j represents a vector 1 km due north. An aircraft flies (at a constant height) with a speed of $800$ km/h. It flies in a fixed ...
-2
votes
1answer
515 views

Volume of a pyramid.

Find the volume of the pyramid with base in the plane $z=−9$ and sides formed by the three planes $y=0$ and $y−x=3$ and $2x+y+z=3$.
1
vote
3answers
177 views

Finding nontrivial solutions to a system of equations

Suppose I have two equations for real numbers $$ a^2+b^2+c^2+d^2+e^2+f^2=1 $$ $$ af-be+cd=0 $$ I would like to find a solution $(a,b,c,d,e,f)$ such that none of its entries are zero and all entries ...
-1
votes
2answers
793 views

Help creating a rational function

Create a rational function with vertical asymptotes $x=\pm1$ and oblique asymptote of $y=2x-3$ and a $y$-intercept of $4$.
6
votes
5answers
117 views

What is the property that allows $5^{2x+1} = 5^2$ to become $2x+1 = 2$?

What is the property that allows $5^{2x+1} = 5^2$ to become $2x+1 = 2$? We just covered this in class, but the teacher didn't explain why we're allowed to do it.
0
votes
3answers
36 views

simplifying $e^\frac{-t}{4} (\frac{-1}{4})+(\frac{-1}{4}t+1) e^\frac{-t}{4}(\frac{-1}{4})$

Im stuck yet once again simplifying a derivative i get so close to finishing a problem then i spend an hour trying to do something that should otherwise be simple. simplifying $$e^\frac{-t}{4} ...
4
votes
7answers
569 views

Proving $\sin (x)=\cos (90^\circ-x)$

I'm interested in the different ways of proving this, any proof is welcome. I understand one way is the cosine sum/difference formula, another is using a right angled triangle. Are there any others? ...
5
votes
4answers
610 views

How to solve $a\cos^2(\theta) + b \sin^2(\theta) = 0$ for $\theta$

How would I solve $$a\cos^2(\theta) + b \sin^2(\theta) = 0$$ for $\theta$. Here $a,b$ are constants and $a \neq b, a \neq 0, b \neq0$. I thought there might not be any solutions but with the ...
1
vote
1answer
52 views

Algebraic functions

I am wondering, if you consider a polynomial in two variables like $$P(x,y)=0,$$ and a zero $P(a,b)=0$ exists fulfilling $P_y(a,b)=0$ and $P_x(a,b)=0$, is there continuity in the sense of $$\forall ...
4
votes
3answers
72 views

How does $3^n-1+2 \cdot 3^n$ evaluate to $3^{n+1}-1$?

Can anybody here help me with this simple problem? I've been thinking about this for half an hour and I am not able to come to a solution. How does $3^n-1+2 \cdot 3^n$ evaluate to $3^{n+1}-1$?
2
votes
1answer
63 views

Proof $ab-cd \ge 3$

$a$, $b$, $c$, and $d$ are real number $a\ge b\ge c\ge d$ $a+b+c+d = 13$ $a^2+b^2+c^2+d^2 =43$ Proof that $ab-cd\ge 3$
1
vote
2answers
71 views

number of roots of polynomial of order n

from theorem of algebra,it is well know that polynomial of order n has exactly n roots,for exmaple quadratic equation like $ax^2+bx+c$ has three cases let $D=b^2-4ac$ ,so we have ...
2
votes
4answers
167 views

Prove the following equation, given that $a,b,c,d$ are reals such that

Given that the following is true for the real numbers $a,b,c,d$ $$\frac{a+b}{c+d}=\frac{a-b}{c-d}$$ Prove $$(a^2 + b^2)(c^2 + d^2)=(ac + bd)^2$$ I cross multiplied the first equation and got ...
2
votes
0answers
37 views

For which minimal $k$ true is that ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$, ${a}_{i}\in {1,2,3,4,5,6}$?

I've got the following inequality, which bounds Minkowski distance. ${4}^{k}\cdot n\leq \displaystyle\sum^{n}_{i=1}{a}_{i}^{k}\leq {5}^{k}\cdot n$ and values of ${a}_{i}\in {1,2,3,4,5,6}$ We know all ...
1
vote
2answers
39 views

What would be cost of toaster in this problem?

A man sold a toaster at a loss of 8%. Had he bought it at 10% less and sold for $88 more he would have gained 20%. What would be cost of toaster.
6
votes
3answers
287 views

How many solutions are possible to this equation?

Given $$A+2B+3C=N $$ where $N$ is a given positive integer. $A ,B,C\in\mathbb{N}$ vary from $0$ to $\infty$. How many solutions will be there to this equation?
3
votes
7answers
735 views

How do I show $\sin^2(x+y)−\sin^2(x−y)≡\sin(2x)\sin(2y)$?

I really don't know where I'm going wrong, I use the sum to product formula but always end up far from $\sin(2x)\sin(2y)$. Any help is appreciated, thanks.
0
votes
3answers
486 views

Solve this: $\sqrt{x^2+3x+6}+\sqrt{2x^2-1}=3x+1$

Solve this: $$\sqrt{x^2+3x+6}+\sqrt{2x^2-1}=3x+1$$ I can solve this by wolfram but I need nice solution
7
votes
4answers
268 views

finding the real values of $x$ such that : $x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$

How to find the real values of $x$ such that : $$x=\sqrt{2+\sqrt{2-\sqrt{2+x}}}$$
1
vote
4answers
246 views

Simultaneous Equations year 9 please solve

$$xy=4\tag1$$ $$2x - y - 7 = 0\tag2$$ Simultaneous Equations Please Solve in in year 9.
10
votes
4answers
244 views

prove that $\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} , \ x>0$,

Can you show very easy methods? I hope I'll see many methods. Thank you everyone. Prove that: $$\frac{1-e^{-x^2}}{x}\le 2\sqrt{2} \ \ \ \qquad \forall x>0.$$
4
votes
3answers
120 views

Use the definition of the derivative for this question.

Differentiate the function f(x)=x^3 in the point a. Use the definition of the derivative for this question. I know that the definition of the derivative is: $$f'(x) = \lim_{h\to ...
0
votes
1answer
98 views

Fractional Surds - Simplifying and Rational Denominator

Simplify $\frac{3}{\sqrt5 + 2} - \frac{\sqrt2}{2.\sqrt2 - 1}$, writing your answer with a rational denominator. So i have solved questions like this in my whole life. But i'm just confused can ...
1
vote
0answers
320 views

Analytical solution for a variable inside of a summation

I am trying to figure out how to solve the following expression for $x$ and I'm surprised that I don't know what to do. $$\frac{2n}{x} = \sum_{i=1}^{n} \frac{1}{x-y_{i}}$$ We have that $n$ and ...
0
votes
2answers
107 views

Find $C$ if $B(B-C)=23$. $B$ and $C$ are positive integers.

I don't know how to tackle the problem. I tried to factor the equation and use systems of equations but it still does not work. Please give a good proof.
2
votes
3answers
71 views

Show that $(n - 1)2^{n+1} + 2 + (n+1)2^{n+1} = n(2^{n+2})+2$

I'm having a really hard time showing this equality is true, I've tried several ways of going about it and I just can't seem to make it work. Help! $(n - 1)2^{n+1} + 2 + (n+1)2^{n+1} = n(2^{n+2})+2$ ...
3
votes
4answers
100 views

Can $|-x^2| < 1 $ imply that $-1<x<1$?

Can $|-x^2 | < 1 $ imply that $-1<x<1$? My steps are as follows? $$| -x^2| < 1 $$ $$-1<(-x^2)< 1 $$ $$-1<(-x^2)< 1 $$ $$-1<x^2< 1 $$ $$\sqrt{-1}<x< \sqrt 1 $$ ...
0
votes
1answer
55 views

Factorising and limits

How do I factorize this expression? $$(2^n-3^n+n4^n)^{\frac{1}{n}}$$ so far I have: $$n4^n\left(\frac{1}{n} \left(\frac{1}{2}\right)^n-\frac{1}{n}\left(\frac{3}{4}\right)^n +1\right)^{\frac{1}{n}}$$ ...
0
votes
1answer
70 views

Solving a non-linear equation of 3 unknowns

I have the following set of equation which comes from a free boundary problem. I have seen a paper which had complicated expression for the solution for even more variables. I would like to know how ...
6
votes
4answers
476 views

How to solve equations like $8^{2n+1} = 32^{n+1}$

I have stumbled on this question, and there are a few questions after it of the same type. How do I solve it and what is the right approach for this kind of question? $$8^{2n+1} = 32^{n+1}$$ I need ...
2
votes
2answers
41 views

why constant derivatives?

Really simple question here. Say $f(x)$ and $g(y)$ then why if $\frac{d f(x)}{dx} = \frac{d g(y)}{dy}$ then both derivatives are constant? Thank you all very much
11
votes
9answers
2k views

7th class question my daughter asked and need answer if possible

My daughter has asked me to solve this question but I am unable to than I thought to post it here may be someone help. Q : The watchman works 4 days a week and has a rest on the fifth day. He had ...
4
votes
3answers
285 views

The matrix I mentioned below is irreducible and primitive or isn't?

$$ \left[ \begin{array}{@{}ccccc@{}} 0.9& 0.1& 0& 0& 0& 0& \\ 0& 0.9& 0.1& 0& 0& 0& \\ 0& 0& 0.9& 0& 0& 0.1& \\ 0& 0& ...
12
votes
1answer
226 views

Why is integer approximation of a function interesting?

I have recently learnt the following result: Let $g \in \mathbb{R}[x]$ be a polynomial with $g(0) = 0$. Then, for any $\varepsilon > 0$, the set of positive integers $n$, such that $g(n)$ is ...
1
vote
1answer
251 views

Establishing formula from recurrence

Can anyone tell me how do we establish a formula from a given recurrence relation? Take the example of $f(n) = 2f(n-1) + 1$, $n \in \mathbb{Z^+}$, $f(1) = 1$ When I write out the first few values, it ...
1
vote
3answers
67 views

Logarithm simplification

Simplify: $\log_4(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}})$ Can we use the formula to solve this: $\sqrt{a+\sqrt{b}}= \sqrt{\frac{{a+\sqrt{a^2-b}}}{2}}$ Therefore first term will become: ...
1
vote
1answer
252 views

Stuck on solving for x in exponential to find variance

The problem seems simple: Let X be an exponential random variable such that $P(X \le 2) = 2P(X > 4)$. Find the variance of X. Easy, right? $ P(x \le 2) = 1 - e^{-2\lambda} $ and $ P(x > 4) = ...
5
votes
5answers
61 views

How to solve simultaneous two variable system

My maths knowledge is rusty and need some help in brushing it up. I tried to google around could not get what i am looking for How to solve the below equation $x + y = 6$ and $x^2 + y^2 = 20$ Help ...
2
votes
2answers
71 views

Knowing when to use factoring/how to factor/rational zeroes

How do I know where I would be using factoring as opposed to rational zero theorem? Do I do Descartes rule of signs to get how many positive/negative and then attempt RZT to get rational zeroes, then ...
4
votes
3answers
233 views

Solving this equation $10\sin^2θ−4\sinθ−5=0$ for $0 ≤ θ<360°$

The first part of the question asks me to square both sides of the equation: $$3 \cos θ=2 − \sin θ$$ So that I can obtain and solve the quadratic: $$10\sin^2θ−4\sinθ−5=0 \;\;\text{for}\;\; 0 ≤ ...
2
votes
4answers
798 views

Explain why there's no solution to the equation $2x-2x^2 = 1$

How can you tell there's no solution to the equation $2x - 2x^2 = 1$. The supporting information goes like this: The diagram above shows the graph of $y = 7 + 2x - 2x^2$. I tried to do this: $2 ...
6
votes
5answers
390 views

Simplify $ \frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}} $

Please help me find the sum $$ \frac{1}{x-y}+\frac{1}{x+y}+\frac{2x}{x^2+y^2}+\frac{4x^3}{x^4+y^4}+\frac{8x^7}{x^8+y^8}+\frac{16x^{15}}{x^{16}+y^{16}} $$
1
vote
2answers
165 views

Show $ \frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1 $ given $xyz = 1$

Please help me prove the equality: If $xyz=1$, prove that $$ \frac{x}{xy+x+1}+\frac{y}{yz+y+1}+\frac{z}{zx+z+1}=1 $$
3
votes
4answers
121 views

How to approach solving this indices example?

I am having problems working out how to approach solving this problem : $$\left(\frac{81}{16}\right)^n = \frac{32}{243}$$ How do I go about working out $^n$? Please step by step if possible.
3
votes
3answers
58 views

How to get from $1 + (-1)^{n+1}$ to $1 + [((−1)^{n }) − 1] (−1)$

I need some help with the algebra here. I have the following explanation, and I really can't follow the algebra. Could you also maybe give me some tips on how to think about such problems. $a_n = 1 ...
8
votes
2answers
112 views

How to solve $4^x - 2^{x + 1} = 3 $ for x?

We figured that this can be changed to $2^{2x} - 2^x \cdot 2 = 3$, but couldn't solve from there. Perhaps we are not on the right path?
3
votes
3answers
1k views

How do I find the maximum and minimum values of $1−4\cos(2\theta)+3\sin(2\theta)$?

To find the maximum of $$1 - 4\cos(2\theta) + 3\sin(2\theta) $$I tried: $$1-4(1)+3(1)=0.$$ To find the minimum I tried to substitute with the minimum values of sin and cos: $$1-4(-1)+3(-1)=2$$ I know ...
0
votes
3answers
152 views

Is this equation solvable for x?

Is is possible to solve this equation for x? $$ \frac{x}{q} + \frac{x}{(1-xk)^2} - t = 0 $$ x, q, k, t are all Real and positive.