1
vote
1answer
121 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
1
vote
3answers
54 views

If one number is thrice the other and their sum is $16$, find the numbers

If one number is thrice the other and their sum is $16$, find the numbers. I tried, Let the first number be $x$ and the second number be $y$ Acc. to question $$ \begin{align} x&=3y &\iff ...
1
vote
3answers
72 views

How does $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ simplify to $1 - \sqrt 2 $?

I've the answer for a question in my textbook to be: $-\sqrt {\frac{{2 - \sqrt 2 }}{{2 + \sqrt 2 }}} $ which i've then simplifed to: $-\sqrt {3 - 2\sqrt 2 } $ However my textbook states $-\sqrt ...
0
votes
2answers
56 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
1
vote
2answers
21 views

Problems with Simplifying Using Factoring of Binomial Expressions

I am running into problems simplifying using factoring of binomial expressions. The problem at hand is this: $(x-1)^3*(2x-3)-(2x+12)*(x-1)^2$ I first expanded the left side of the minus sign, like ...
1
vote
1answer
30 views

solving system of equations(nonlinear)

I am trying to solve the following system of equations: $$\frac{kq^2}{d}=mg(L-L\cos(t))+\frac{kq^2}{r}$$ $$\sin(t)=\frac{x}{L}$$ $$r^2=(L-L\cos(t))^2+(x+d)^2$$ The parameters are: $k,L,d,q,m,g$ The ...
0
votes
1answer
62 views

solving the equation

let there be a function $ f(x)= \ln x-kx^2, k>0$ determine for whihc values of $ k$ ,the equation $f(x)=0.5$ has a single solution; attemp to solve: $$0.5 = \ln x-kx^2$$ $$kx^2 +0.5 = \ln x $$ ...
1
vote
4answers
67 views

inverse trigonometric equation $\displaystyle \tan^{-1}{x}+\cot^{-1}{x}=\frac{\pi}{2}$

I have problem with showing that $\displaystyle \tan^{-1}{x}+\cot^{-1}{x}=\frac{\pi}{2}$ I think there have to be used formula: $\displaystyle ...
2
votes
2answers
133 views

Solve for $x$: $\frac1e = e^{2x}$

I tried making it to $e^{-1} = e^{2x}$ and had the exponents equal each other $-1=2x$ and the I solved for $x$, making it $x=-1/2$, but that answer is wrong. please help I don't know why that ...
1
vote
1answer
36 views

How would I solve these types of equations

Going back to college and been a few years since I've had to do any algebra/trig. How would I go about solving these types of equations and do they have a name? a(y-b)=by+c then, except when the ...
3
votes
1answer
88 views

A unfamiliar question

I'm sure asking this kinda problem is stupid but somehow I have never seen such problems before. $2{x}^2 + 3{y}^2 =0$ what is $3x+2y$?
13
votes
2answers
503 views

Intuitive ways to get formula of cubic sum

Is there an intuitive way to get cubic sum? From this post: combination of quadratic and cubic series and Wikipedia: Faulhaber formula, I get $$1^3 + 2^3 + \dots + n^3 = \frac{n^2(n+1)^2}{4}$$ I think ...
1
vote
4answers
55 views

Trigonometric functions of the acute angle

Find the other five trigonometric functions of the acute angle A, given that: \begin{align} &\text{a)}\ \ \sec A = 2 \\[15pt] &\text{b)}\ \ \cos A = \frac{m^2 - n^2}{m^2 + n^2} \end{align} ...
2
votes
4answers
37 views

Number of distinct real roots with $e^{-x}$ in the equation

How to find the number of distinct real roots of the equation $$13x^{13}-e^{-x}-1=0$$ I know that we generally find number of real roots by observing number of sign changes in $f(x)$ and $f(-x)$ but ...
4
votes
1answer
17 views

Solving Differential equations with Laplace transform

$\displaystyle y''+4y'+3y=e^{-t}$, given $\displaystyle y(0)=y'(0)=1$ My Attempt: Taking Laplace transforms on both sides $\displaystyle $ $\displaystyle [s^2\bar y-sy(0)-y'(0)]+4[s\bar ...
1
vote
1answer
25 views

rearrange $t - (m-q)^2 = v - (m-p)^2$ for quadratic formula form $ax^2 + bx +c = 0$ solving for $q$

I have the equation $t - (m-q)^2 = v - (m-p)^2$ which I would like to rearrange to be able to apply the quadratic formula, and solve in terms of $q$. Accordingly, it needs to be in the form: $ax^2 ...
1
vote
3answers
37 views

solve $-(x_m - x_q)^2 = -(x_m - x_p)^2$ in terms of $x_q$

I have an equation, $-(x_m - x_q)^2 = -(x_m - x_p)^2$ which I want to solve in terms of $x_q$. I can see (by using a number line) that $q$ can have two solutions: $x_q = x_p$ or: $x_q = 2x_m-x_p$ ...
0
votes
4answers
76 views

Questions about solving inequality: $2 < \frac{3x+1}{2x+4}$

Solve the inequality: $2 < \frac{3x+1}{2x+4}$ Step 1: I simplified $\frac{3x+1}{2x+4}$ into: $3x+1-2x-4= x-3$. Step 2: $2>x-3$ Here I subtracted $2$ from both sides into: $x>-5$ or ...
1
vote
2answers
52 views

Inequality - Find what value of $t$ satisfies: $ (t/24) - (t+1) + (3t/8) < (5/12) (t+1)$

Inequality - Find what value of $t$ satisfies: $(t/24) - (t+1) + (3t/8) < (5/12) (t+1)$. Step 1: I multiplied both sides by $24$ and divided to get: $t-24(t+1)+9t < 10+24(t+1)$. Step 2: I ...
1
vote
4answers
1k views

What's wrong with my aproach to solving this equation with multiple logarithms?

A question I was faced with asked "For which $x$ is $\log_{10}(x)^{\log_{10}(\log_{10}(x))}= 10,000$?" My instincts tell me I can say $$\log_{10}(x)=10$$ and $$\log_{10}(\log_{10}(x))=4$$ However, ...
2
votes
1answer
44 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
1
vote
1answer
41 views

Find value of $x$ for: $(1/3)(1-x) \geq 2(x-3)$

Find what value of $x$ satisfy: $(1/3)(1-x) \geq 2(x-3)$ First I multiplied both sides by $3$ so that $1/3$ became $3/3=1$. So I tried to find $x$ this way: $(1-x) \geq 6(x-3)$. I tried solving it ...
1
vote
0answers
18 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
2
votes
3answers
140 views

Deriving the sum-to-product identities

I've been asked by my textbook to derive the "sum-to-product" identities from the "product-to-sum" identities. I've attempted to to do this but i've met a dead end, and i'm quite confused. Using ...
2
votes
3answers
88 views

$\ \sqrt{x+39}-\sqrt{x+7}=4 $

So I tried to solve this problem for x $\ \sqrt{x+39}-\sqrt{x+7}=4 $ I multiplied both sides ($\ \sqrt{m}\cdot\sqrt{n}=\sqrt{mn} $) $\ (\sqrt{x+39}-\sqrt{x+7})^2=16 $ $\ ...
0
votes
3answers
53 views

sketching the graph

how to sketch the graph $|x+y|= m$, when $ m $ is some real number? I personally can not see any efficient or known method to do so...
0
votes
4answers
54 views

solving the system

solve the system : $$ y+|x-2|=3 $$, $$ |x+y|= m $$ graphicly when $m$ equals $6$. I can easily (realtively) skecth the first graph , however, how the bloody hell do you sketch $|x+y|= 6$??
0
votes
2answers
24 views

Trigonometry Identities questions

Given that $\sin\theta =\dfrac15$ and $0<\theta <\dfrac{\pi}2$, without evaluating the angle $\theta$, find the exact value of $$\sin\left( \frac{\theta}2-\theta \right)\tag1$$ I know that ...
3
votes
5answers
192 views

Calculating the area

For the two graphs $ \frac{x^3+2x^2-8x+6}{x+4} $ and $ \frac{x^3+x^2-10x+9}{x+4} $, calculate the area which is confined by them; Attempt to solve: Limits of the integral are $1$ and $-3$, so I took ...
0
votes
1answer
24 views

Calculating optimum values of $u$ and $m$ from $\mathbb V(\bar {y_2}\prime)=\frac{S_2^2(n-u\rho^2)}{n^2-u^2\rho^2}$

I have to find optimum sample size in sampling on two occasions. Suppose that the samples are of the same size n on both occasions. In selecting the second sample, $m$ of the units in the first ...
0
votes
2answers
37 views

Indices: Negatives and fractions?

I need to rewrite this expression using powers - Rewrite these expressions as a power of the given number: I know that a reciprocal of a power is when it halves. e.g. $2^0$ $=1$ $2^{-1}$ $= ...
-1
votes
1answer
42 views

How many coloured pencils? [closed]

Drawing pencils cost $8$ cents each and coloured pencil cost $11$ cents each. Two dozen assorted pencils cost $\$2.16$. How many coloured pencils are there?
1
vote
1answer
40 views

solving logarithmic equations by expressing in terms of exponents

Is it always valid to solve logarithmic equations by raising both sides as powers of a common base? As in: $$ln(x) = ln(y) $$$$ e^{ln(x)} = e^{ln(y)} $$$$ x = y $$$$ where \quad x,y ∈ ...
8
votes
4answers
249 views

$x<y$ then $x^3<y^3$

I'm looking for a proof to the following theorem: For any $x,y\in R$: $x<y \Rightarrow x^3<y^3$ I'm trying this approach: Let $z = x^3 - y^3 = (x-y)(x^2+xy+y^2) = z_1 z_2$ where $z_1 ...
3
votes
1answer
45 views

For which angles is inequality true

My problem is from Israel Gelfand's Trigonometry textbook. Page 48. Exercise 6: a) For which angles $\alpha$ is $\sin^4\alpha-\cos^4\alpha > \sin^2\alpha-\cos^2\alpha$ ? b) For which angles ...
0
votes
1answer
38 views

Basic Mixture question

Let's say that I have 50 ounces of 100% sugar water solution. If I add 50 more ounces of pure water to that solution. What percentage of sugar water solution would I have left?
0
votes
1answer
29 views

can you help me solve my menu board dilemma?

If I have a menu board that measures 35 3/8 $\times$ 71 5/8 and I need to cut in 3 equal pieces, what measurements should each piece be?
3
votes
2answers
80 views

Simplify expression $(x\sqrt{y}- y\sqrt{x})/(x\sqrt{y} + y\sqrt{x})$

I'm stuck at the expression: $\displaystyle \frac{x\sqrt{y} -y\sqrt{x}}{x\sqrt{y} + y\sqrt{x}}$. I need to simplify the expression (by making the denominators rational) and this is what I did: ...
0
votes
1answer
41 views

Vector calculation question…

in the formula for the calulation of the angle between 2 vectors $$\cos \theta \overset{\text{def}}= \dfrac{\vec\alpha \cdot \vec\beta}{|\vec \alpha|\cdot |\vec \beta|}$$ is the output angle is ...
0
votes
3answers
43 views

Solving the complex polynomial

For the complex polynomial $z^3 -5z^2 +(7-2i)z +6i-3 = 0 $ $1)$ show that $2+i $ is a root. $2)$ solve the given equation. Attemp to solve: I'm not really sure how to solve this, but I ...
0
votes
2answers
27 views

Applying the cosine even identity to the cosine difference identity

I'm slightly confused over what happens when you're applying cosine's "even identities" to the difference identity. Here's how I go about, please tell correct me as I feel i'm going wrong somewhere. ...
0
votes
1answer
65 views

A few questions regarding the cosine difference identity

I've a few questions that stem from the proof given in my textbook regarding the cosine difference identity. The proof goes like this: Let $\alpha$ and $\beta$ be angles plotted in standard ...
0
votes
0answers
20 views

How to evaluate the graph? [duplicate]

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
1
vote
1answer
72 views

How to solve this graphing question?

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
1
vote
2answers
49 views

How to express a trigonometic equation in $\sin 2\theta $ and $\cos 2\theta $?

How do I express the given equation in $\sin 2\theta $ and $\cos 2\theta $ in terms of x? $x + 3 = 7\sin \theta $ with $\frac{\pi }{2}{\text{ < }}\theta {\text{ < }}\pi $ for $\sin 2\theta ...
0
votes
2answers
39 views

Solving equations with powers without logarithms

Im taking an introduction to logarithms. Of course a short review of exponentiation is inherent for a clear understanding of logarithms. I was asked to find, for example, $27^x = 3$. (without the use ...
1
vote
1answer
35 views

Formula alteration

is there any way to transform the formula$ \frac {1-x}{x-3}$ into something that can be easily sketched, or which will help eliminate $x$ from the denominator?
1
vote
1answer
43 views

Completing the following equation by the suitable method

i got this linear equation two variable problems for my school. I understand the basics of the normal linear equation but this seems different instead having a pure number after the "=" they got a ...
0
votes
2answers
70 views

Primary school mathematics regarding age

Mrs Lim is 40 years old and her son is twice her daughter's age. Mrs Lim will be thrice her son's age when her daughter is 12 years old. How old will mrs Lim be when her 2 children's combined age ...
1
vote
2answers
103 views

Omitting $i$ in calculations

Is it possible in various calculations related to the complex plane which also include analytic geometry , calculating distances etc, to omit $i$ and treat the imaginary axis as simply the cartesian ...