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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8
votes
3answers
562 views

If two parabolas don't intersect, is there a line that doesn't intersect either of them?

Two parabolas in a plane are given, such that they don't intersect. Is it true that there is a line in plane such that doesn't intersect any of them?
-3
votes
3answers
964 views

Simplifying $\sqrt {1+(x/2 - 1/(2x))^2}$

I am having trouble figuring this out. $$\sqrt {1+\left(\frac{x}{2}- \frac{1}{2x}\right)^2}$$ I know that $$\left(\frac{x}{2} - \frac{1}{2x}\right)^2=\frac{x^2}{4} - \frac{1}{2} + \frac{1}{4x^2}$$ ...
1
vote
1answer
80 views

Does this inequality hold in general?

If I have functions $f,g,h > 0$ and $f\le g+h$ then: $$\frac f{f+1}\le \frac g{g+1}+\frac h{h+1}, x \in R$$ I have been trying to find out whether it's true or not but I haven't succeeded.
4
votes
2answers
324 views

How can we solve: $\sqrt{x} - \ln(x) -1 = 0$?

How could we solve $$\sqrt{x} - \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions? This question is inspired by my first question How ...
3
votes
2answers
584 views

How can we solve: $\sqrt{x} + \ln(x) -1 = 0$?

How could we solve $$\sqrt{x} + \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions?
3
votes
4answers
2k views

Polar equation of a circle

A very long time ago in algebra/trig class we did polar equation of a circle where $r = 2a\cos\theta + 2b\sin\theta$ Now I forgot how to derive this. So I tried using the standard form of a circle. ...
2
votes
5answers
323 views

$3x^3 = 24$ quadratic equation

Completing the square I know by factoring $$x^3 - 8 = 0\\ x-2 = 0$$ that one of the solutions is 2. but the other solutions is $1 ± i \sqrt 3$. Can someone explain to me how to get that?
4
votes
1answer
162 views

quadratic inequality

I do a procedure for solving algebraic inequalities of the second ($x^2+bx+c>0$) degree for my student. I know it is possible to solve the inequality by factorisation, Solving a quadratic ...
2
votes
3answers
412 views

Solve an absolute value equation simultaneously

My question is : Solve simultaneously $$\left\{\begin{align*}&|x-1|-|y-2|=1\\&y = 3-|x-1|\end{align*}\right.$$ What I did : $y=3 - |x-1|$ is given. Thus $y = 3-(x-1)$ or $y = ...
2
votes
3answers
149 views

Where is $f(x)={(x^2+2x-48)}/{x^2}$ increasing? decreasing?

The question is to find where the graph is increasing or decreasing. The original function is $f(x)={(x^2+2x-48)}/{x^2}$ I know I need to find the prime of this function and I think it is this ...
0
votes
3answers
3k views

Long division in integration by partial fractions

I am trying to figure out what my book did, I can't make sense of the example. "Since the degree of the numberator is greater than the degree of the denominator, we first perform the long division. ...
0
votes
2answers
107 views

about solving: Absolute value

How to solve: $|\sqrt{x-1}-2| + |\sqrt{x-1}-3|=1$. I would like to know how to solve an absolute value equation when there is a square root sign inside.
2
votes
4answers
370 views

e's with x in the exponents: Gotta solve for x.

I usually get lost when there are these exponential questions. I'm not used to seeing them. I must solve for x. $$f(x)=e^{0.5x}+324e^{-0.5x}=0$$ If I do $f(x)=\ln e^{0.5x}+\ln 324e^{-0.5x}=\ln 0$ ...
1
vote
2answers
98 views

Function transformation

I have the function $f(x)=-\frac{1}{x}$, plotted on standard 2D axes. I take the vertical asymptote and rotate it $45$ degrees clockwise, so that it's now $y=x$ instead of the $y$-axis. The ...
4
votes
2answers
579 views

Solving $\sqrt{x^2 +2x + 1}-\sqrt{x^2-4x+4}=3$

My question is: Solve $\sqrt{x^2 +2x + 1}-\sqrt{x^2-4x+4}=3$ I deduced that:$LHS= x+1-(x-2)$ I am unable to solve this equation. I would like to get some hints to solve it.
1
vote
2answers
155 views

Solving $|x-2| + |x-5|=3$ [duplicate]

Possible Duplicate: How could we solve $x$, in $|x+1|-|1-x|=2$? How should I solve: $|x-2| + |x-5|=3$ Please suggest a way that I could use in other problems of this genre too Any help ...
0
votes
2answers
66 views

Solving for $x$ in terms of $y$

For some reason, I'm having a hard time solving for $x$ in this equation: $$x^2=y,-2\lt x \lt 3.$$ I could use some help. Thanks.
2
votes
5answers
337 views

question about real numbers

My question is: Solve $(x-a)(x-b)(x-c)=0$ where $a,b,c$ belong to real numbers. By observing, I found out that $x$ can be $a$ or $b$ or $c$.
-1
votes
3answers
150 views

An inequality with absolute value and a parameter: $|x-4|>a$

Solve : $|x-4|>a$. Case 1: $a>0$; Case 2: $a<0$ Progress I am getting answers which look similar in both cases: Let $a>0$ so $x>4+a$ or $x<4-a$ , Let $a<0$ so ...
1
vote
1answer
198 views

Range Formula Between Two Values?

I'm in the middle of programming something, and I really need a formula that can go from 0 to 1 depending on the distance an object is to another one. I can get the formula to return 1 if the objects' ...
0
votes
1answer
648 views

Intersection point of tangent line with $X$ axis

i have confused in one topic and please help me,suppose that we have following function $f(x)=x^3+x^2-2*x-3)$ we know that there is a tangent of this function which goes through point $(1,-3)$,we ...
3
votes
2answers
169 views

Summation of $ \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \cdots$ till $n$ terms

What is the pattern in the following? Sum to $n$ terms of the series: $$ \frac{1}{2} + \frac{3}{4} + \frac{7}{8} + \frac{15}{16} + \cdots$$
0
votes
2answers
211 views

Finding a point on the line joining two points in Cartesian coordinate system

Given a point A with coordinates (a,b) ; a point B with coordinates (c,d) , I want to find a point C's coordinate (x,y) ,with C lying on the line joining A and B and C is at a distance Dist from point ...
1
vote
1answer
570 views

Formula for calculating rate drain

A question which I am having difficulty with is: A tank is 3/4 full. Fillpipe A can fill the tank in 12 min. Drainpipe B can empty the tank in 8 min. If both pipes are open, how long will it take ...
0
votes
1answer
52 views

Algebra factor question?

Can anyone help me factor this equation? (One hundred sixty)(one minus q divided by four)(ten plus q divided by 4)
2
votes
1answer
101 views

How do you solve this equation: $5x^{\frac{1}{x}}=3$

I've been at it for a while but I can't get it. Can anyone help?
1
vote
1answer
175 views

Absolute value of a real number

My question is: Solve: $|x-4|< a$, where $a$ belongs to the real numbers. Solve this by considering various cases depending upon whether $a$ is negative, positive or zero. What I have tried ...
0
votes
1answer
192 views

Simultaneous equations

My question is: Solve simultaneously:(anwers are in integers) $$\begin{align} y^3 - 9x^2 + 27x - 27 &= 0 \\ z^3-9y^2+27y-27 &= 0 \\ x^3-9z^2+27z-27 &= 0 \end{align}$$ Any hints to solve ...
1
vote
2answers
566 views

Quadratic equation related to physics problem - how to proceed?

It's a physics-related problem, but it has a nasty equation: Let the speed of sound be $340\dfrac{m}{s}$, then let a heavy stone fall into the well. How deep is the well when you hear the impact ...
2
votes
0answers
224 views

Getting an upper bound for a function of two variables

I have a complicated looking expression involving two naturals numbers $n,k$ where $n\ge k\ge 3$: ...
0
votes
2answers
77 views

system of simultaneous equations

My question is: Solve simultaneously: $$\left\{\begin{align*} &\frac{xy}{x+y}=1\\ &\frac{xz}{x+z}=2\\ &\frac{yz}{y+z}=3 \end{align*}\right.$$ I am unable to solve this ...
4
votes
2answers
130 views

How to find $A_1A_2 + \cdots + A_{2010} A_{2011}$, where $A_{n+1} = \frac{1}{1+\frac{1}{A_n}}$

My question is: If $$A_{n+1} = \frac{1}{1+\frac{1}{A_n}}$$ ($n\in\mathbb{N}$) and $A_1=1$, then find the value of: $$A_1A_2 + A_2A_3 + A_3A_4 + \cdots + A_{2010} A_{2011}.$$ Please I would ...
3
votes
1answer
185 views

Solving $\sqrt[3]{x^2} + \sqrt[3]{x} = 2$

My sister asked me for some help on her algebra homework the other day, and I was stumped by her question. The problem is to find the root of $\sqrt[3]{x^2} + \sqrt[3]{x} = 2$. The internet tells me ...
0
votes
1answer
77 views

Related to linear equation with one unknown

My question is: Given that the equation $\,\frac{8}{3}x - a = \frac{9}{4}x + 123\,$ has positive integral solution where a is also positive integer, find the minimum possible value of a. Please any ...
0
votes
1answer
54 views

for Y(b) = thing1, after a transform Y(a)=thing2 , is there Y(a,b)= thing3?

For $Y(b) = \text{thing}_1$, after a transform to another domain $Y(a)=\text{thing}_2$ , is there $Y(a,b)= \text{thing}_3$? where $\text{thing}_3$ is related to $\text{thing}_1$ and $\text{thing}_2$ ...
3
votes
2answers
387 views

Proving $\sin x + \sin x \cdot \cot^2 x = \csc x $

The exercise is to prove the trig identity by rewriting each side of the equation into the same form. However only the following identities can be used in the process: $$\begin{align*} \tan \theta ...
0
votes
1answer
54 views

Related to linear equation with one variable

My question is- Solve completely: $$ ax + b - \frac{5x + 2ab}{5} = \frac{1}{4}$$ Any guidance to solve this question would be helpful.
6
votes
2answers
186 views

System of two Equations

A friend of Mine gave me a system of two equations and asked me to solve them $\rightarrow$ $$\sqrt{x}+y=11~~ ...1$$ $$\sqrt{y}+x=7~~ ...2$$ I tried to solve them manually and got this horrendously ...
0
votes
2answers
97 views

Solving $11x-2=kx+15$ for a positive integral solution for $x$ given that $k$ is an integer

My question is- Find the integer value of $k$ such that the equation $11x-2=kx+15$ has positive integral solution for $x$. Find that solution. Any guidance to solve this question would be helpful.
1
vote
3answers
146 views

linear equation in one variable

My question is - Find the values of $a,b$ if the equation $a(2x+3) + 3bx=12x+5$ has infinitely many solutions. Please can anyone guide me to solve these types of questions? I would be really ...
0
votes
2answers
85 views

Is anyone familiar with how this method works? (Uses diagram to solve a word problem)

I remember back in school (some time ago) we were taught to solve problems such as the following: If $6$ men can do $1/3$ of work in $10$ days then how many days would it take $4$ men to do $2/3$ ...
1
vote
1answer
531 views

Law of Cosines Distance Formula Proof

So I'm trying to understand a law of cosines proof that involves the distance formula and I'm having trouble. I've included the proof below from wikipedia that I'm trying to follow. What I'm have ...
0
votes
2answers
206 views

How long does it take these boys to paint a fence?

I came across the following question: If 3 boys can paint a fence in 2 days what part of the job can be completed by two boys in 1 day? The answer to this problem is $\frac13$. I can't ...
1
vote
4answers
361 views

Find x when the function equals 0

I must solve for x for this function. $e^x-20x=0$ I'm not sure what to do here. I've tried this so far but it makes no sense: $$\begin{align*} e^x&=20x\\ x\ln e&=\ln20+\ln x\\ ...
0
votes
1answer
169 views

Parameter or independent variable?

I need an explanation of the difference between parameter and variable in the following example. In extremal geometric problems when we want to find the object having some extremal property, say ...
0
votes
3answers
80 views

Algebra simplify question?

How would I simplify this? $$5\% \cdot \frac12 \left(3000 + 2x\right)$$
0
votes
2answers
76 views

A matrix's element proof

Thanks again for copper.hat and Robert Israel's quick immediate reply. While I am modifying the questions, they've already given the answer. Now in this thread, I've changed it back to the original ...
0
votes
1answer
129 views

Simplify this algebraic fraction

I have this algebraic fraction: $$\frac{t^4-1}{t^2-t^6}$$ And I'm told the answer is: $$\frac{-1}{t^2}$$ I can't for the life of me work out how to simplify it. (I'm sorry for the simple question) ...
0
votes
1answer
64 views

what is general theory for those type of problem: To think about the condition must be exist if a decimal with infinite digit in base 10 …

what is general theory for those type of problem: To think about the condition must be exist if a decimal with infinite digit in base 10 also have infinite digit in both base 3, base 4? please prove ...
3
votes
4answers
192 views

How to solve $y = \ln\frac{1+x}{1-x}$?

Can the following equation be solved for x? $$ y= \ln\frac{1+x}{1-x} $$ *This is not actually homework - it comes from part of the rate law for a particular chemical reaction I am studying - but ...