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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1
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1answer
417 views

Variation of parameters formula with complex imaginary roots

I am needing to use the Variation of parameters formula to solve a second order non-homogeneous equation. I have used this before however i now have an equation with complex imaginary roots My second ...
1
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4answers
96 views
0
votes
1answer
38 views

Pre-Algebra Fractional Exponent Question

Why does $t^{\frac{3}{2}} \cdot t^{\frac{1}{2}} = t^2$? What I tried to do was multiply the exponents together $\frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}$ so my final answer was $t^{\frac{3}{4}}$ ...
2
votes
3answers
473 views

Is there a comprehensive algebra exercises site online?

I'm trying to find a place that has guided walkthroughs/answer guides for intermediate algebra, college algebra, and precalc. Maybe trig, also. Any ideas?
1
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2answers
25 views

Algebraic Manipulation

Given that $ a^2 - b^2 = 60 $ and $a - b = 6 $ $a + b = 10$ Find value of $a\cdot b$ I tried $(a-b)^2 = 6^2 \longrightarrow a^2 - 2ab + b^2 = 36$
2
votes
4answers
127 views

Common tangents to circle $x^2+y^2=\frac{1}{2}$ and parabola $y^2=4x$

I'm having trouble with this. What i do is say $\epsilon: y=mx+b$ is the tangent and it meets the circle at $M_1(x_1,y_1)$, i equate the $y$ of the tangent with the circle: $y=\pm \sqrt{1/2-x^2}$ and ...
2
votes
3answers
72 views

How to find range of $\frac{\sqrt{1+2x^2}}{1+x^2}$?

How to find range of $$\frac{\sqrt{1+2x^2}}{1+x^2}$$ ? I tried put it equal to $y$ and squaring but I'm getting $4$th degree equation.
4
votes
2answers
109 views

A subset of easily solved 4th degree polynomials

I've found (maybe, maybe not, but it's not on this Wikipedia or this Wikipedia) that there is a subset of easily solved quartic polynomials of the form $$0=x^4+2px^3+(p^2+p+2q)x^2+(2pq+p^2-1)x^1+(q^2+...
1
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4answers
60 views

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$?

How to solve $|-2x^2+1+e^x+\sin(x)|=|2x^2-1|+e^x+|\sin(x)|$ ? I've solved equations like $|a|+|b|=|a+b|$ where the condition must be that $a$, $b$ must be of same sign. But in case of three terms ...
1
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7answers
126 views

$5^{th}$ degree polynomial expression

$p(x)$ is a $5$ degree polynomial such that $p(1)=1,p(2)=1,p(3)=2,p(4)=3,p(5)=5,p(6)=8,$ then $p(7)$ $\bf{My\; Try::}$ Here We can not write the given polynomial as $p(x)=x$ and $p(x)=ax^5+bx^...
1
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1answer
41 views

Given: 2 lines containing the diameter of a circle and a point lying on this circle; Find: the equation of this circle

The lines $ y = \frac{4}{3}x - \frac{5}{3} $ and $ y = \frac{-4}{3}x - \frac{13}{3} $ each contain diameters of a circle. and the point $ (-5, 0) $ is also on that circle. Find the equation of this ...
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votes
0answers
80 views

Given 2 lines containing the diameter of a circle and a point lying on this circle, find the equation of this circle [closed]

The lines $ y = \frac{4}{3}x - \frac{5}{3} $ and $ y = \frac{-4}{3}x - \frac{13}{3} $ each contain diameters of a circle and the point $ (-5, 0) $ is also on that circle. Find the equation of this ...
0
votes
3answers
92 views

Very Basic Math question?

How can I prove that $$\frac{r}{(1-x)^2} + \frac{rx}{x(1-x)} = \frac{r}{x(1-x)^2}$$ I have tried to prove that , but I could not , can someone help me please ? Thanks
2
votes
4answers
110 views

Solution of $(n+1)^{1/3}-n^{1/3}=\frac{1}{12}$

Solve the given equation for $n$ $(n+1)^{1/3}-n^{1/3}=\frac{1}{12}$ How to approach this particular question? Sorry cannot show any work because the only approach I can see is take cube on both ...
1
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2answers
82 views

Solution of given equation for $x$

Solve the given equation for $x$ $$\sqrt{x^2-2x+8}+\sqrt{x^2-2x+3}=125$$ I solved the question by taking ${x^2-2x+3}=t$, and squaring twice and finally solving ${x^2-2x+3}=t$ but it required very ...
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2answers
68 views

Algebra question?

So when I plug in: $$x^4 + \sqrt{3x^2} - 7 = 0$$ I get the roots as: $x = 1.45$ and $x = -1.45$ Somehow that's wrong, can anyone confirm this?
0
votes
1answer
78 views

For what real values of $c$ does $x-\ln {(1+e^x)}=c$ hold for some $x$?

For what real values of $c$ does $x-\ln {(1+e^x)}=c$ hold for some $x$? Rewriting the equation as $\ln {e^x}-\ln {(1+e^x)}=c$, I get $e^c=\frac{e^x}{1+e^x}$. Not sure how to proceed from here.
-1
votes
1answer
39 views

Find $a$ and $b$ if $0 < x < 5$ then $a < x + 2 < b$.

I am trying to solve this simple equation but I am not able to figure out how to do it. Can somebody give me a hand? If $0 < x < 5$ then $a < x + 2 < b$.
0
votes
2answers
62 views

If $\sqrt{18-6\sqrt{5}} = \sqrt{a}- \sqrt{b}$, then which of the following relations are true?

I am stuck at a question. If $\sqrt{18-6\sqrt{5}} = \sqrt{a}- \sqrt{b}$, then which of the following is true: $a+b= 18$ $a+b= 16$ $a+b= 20$ $a-b= 18$ I tried to first take ...
2
votes
3answers
48 views

induction clarification about the step $n+1$

Suppose i need to prove that $\frac{1}{2^2}+\frac{1}{3^2}...+\frac{1}{n^2}<1-\frac{1}{n}$ So in the step of $n+1$, the right side becomes $<1-\frac{1}{n+1}$ or is it: $<1-\frac{1}{n}-\frac{1}...
7
votes
2answers
94 views

How to arrive at Ramanujan's nested radicals?

Ramanujan found that $\sqrt[3]{\cos\left(\frac {2\pi}{7}\right)}+\sqrt[3]{\cos\left(\frac {4\pi}{7}\right)}+\sqrt[3]{\cos\left(\frac {8\pi}{7}\right)}=\sqrt[3]{\frac {1}{2}\left(5-3\sqrt[3]{7}\right)}$...
0
votes
1answer
30 views

Sum algebra solving for coefficient

Is the following equation solvable for $k$? $$\sum_{i=1}^{n}\frac{x_ie^{kx_i}}{1+e^{kx_i}} = 0$$
0
votes
2answers
48 views

Finding $\lim_{L \to \infty} \exp{\frac{T}{L}}\sum_{i=1}^L[ \exp{iA + (i-1)B}]$

I am working on a problem and I have come up with a formula that I would like to simply. WLOG, it looks like the following: $\exp{\frac{T}{L}}\sum_{i=1}^L[ \exp{iA + (i-1)B}]$ Here, $A,B, T$ are ...
5
votes
2answers
51 views

Given any finite string of number, is it true there exists a perfect square whose leading numbers are the string

Given any finite string of number, is it true there exists a perfect square whose leading numbers are the string? For example, given the string 123456, can I find a perfect square with leading digits ...
1
vote
1answer
80 views

Evaluate $\lim_{x \to 0} \frac{\tan(\tan x)-\sin(\sin x)}{\tan x-\sin x}$

Evaluate $$\lim_{x \to 0}\frac{\tan(\tan x)-\sin(\sin x)}{\tan x-\sin x}$$ First I tried using L'Hopital's rule..but it's very lengthy Next I have written the limits as $$L=\lim_{x \to 0}\frac{\tan(\...
1
vote
1answer
57 views

Simple inequality

$$2<\frac{x}{x-1}\leq3$$ What I did is: $$2<\frac{x}{x-1}\leq3\Rightarrow 2<\frac{x-1+1}{x-1}\leq3 \Rightarrow 2<1+\frac{1}{x-1}\leq3\Rightarrow 1<\frac{1}{x-1}\leq 2\Rightarrow$$ $$...
3
votes
3answers
4k views

Find the coordinate of third point of equilateral triangle.

I have two points A and B whose coordinates are $(3,4)$ and $(-2,3)$ The third point is C. We need to calculate its coordinates. I think there will be two possible answers, as the point C could be on ...
0
votes
0answers
19 views

A question based on finding range of a function.

$\mathbf{Question:}$ If $[5\sin(x))] + [\cos(x)] + 6=0$ then what is the range of $f(x)=\sin(x) + \sqrt{3}\cdot \cos(x)$, corresponding to the solution set of the given equation? (Where [.] denotes ...
0
votes
3answers
58 views

How to draw the plane $x+y+2z=2$

I need to evaluate $$\iiint \text d x \text d y\text d z$$ the planes are $x=0,y=0,z=0,x+y+2z=2$ Is there a method to draw the plane? it is easy to draw $x=0,y=0$ and $z=0$ but how can I draw to ...
0
votes
2answers
44 views

|3x-15|<6+|9-9x/5| Find the range of values of x?

How do I solve this modulus inequality? I don't understand the ways to do this question such as squaring and taking out the modulus twice.Can someone explain the ways to do this question?Thanks!
0
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0answers
20 views

separate $a$ & $b$ in $ssrt(a^a*b^b)$

It is already known that $ssrt(a^a*b^b)$ does not equal $ssrt(a^a)*ssrt(b^b) = a*b$ Is there any other method to separate $a$ and $b$?
4
votes
2answers
88 views

problem proving: $(1+q)(1+q^2)(1+q^4)…(1+q^{{2}^{n}}) = \frac{1-q^{{2}^{n+1}}}{1-q}$

I'm trying to prove this, and it is really frustrating, because it seems a really easy problem to prove, however, I'm having a little problem with exponents: $$(1+q)(1+q^2)(1+q^4)...(1+q^{{2}^{n}}) = ...
45
votes
10answers
1k views

What are Different Approaches to Introduce the Elementary Functions?

Motivation We all get familiar with elementary functions in high-school or college. However, as the system of learning is not that much integrated we have learned them in different ways and the ...
1
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5answers
100 views

How to obtain this factorization of $x^4+4$?

$x^4 + 4 = (x^2 + 2x +2)(x^2 - 2x +2)$ I am curious how would one obtain this factorization? Clearly, once the factorization is known it is routine to verify it, however the hard part is how to find ...
2
votes
1answer
70 views

Strange Algebra

I am working on a mathematical induction worksheet and my professor gave us the key and I have run across something that makes zero sense to me so please explain if you can. Additional info: $k \ge2$ ...
4
votes
3answers
115 views

Why is finding the roots of a polynomial equation so important? What is to gain? [duplicate]

I have just started a pre calculus class, and our first lessons have been reviews on polynomial equation, quadratics and finding roots or solutions to equations. The topic is fairly simple but I just ...
1
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1answer
17 views

Prove that $a(x)$ divides $(v(x) - t(x))$

"Let $a(x), b(x) \in \mathbb{R}[x]$, not both the zero polynomial and suppose that gcd[$a(x), b(x)$] = 1. Let $u(x), v(x) \in \mathbb{R}[x]$ be such that $a(x)u(x) + b(x)v(x) = 1$ Let also $s(x)t(x) ...
8
votes
2answers
203 views

Last digit of $3^{459}$. [duplicate]

I am supposed to find the last digit of the number $3^{459}$. Wolfram|Alpha gives me $9969099171305981944912884263593843734515811805621702621829350243852275145577745\\...
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votes
1answer
96 views

Basic Algebra $x^3+y^3=?$ [closed]

$(x^2+1)(y^2+1)+16=8(x+y)$ $x^3+y^3=?$ $x,y\in \mathbb R$ I have nothing something worthwhile. Please give hint.
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3answers
49 views

Help with Trigonometric Functions

so while playing around with circles and triangles I found 2-3 limits to calculate the value of $ \pi $ using the sin, cos and tan functions, I am not posting the formula for obvious reasons. My ...
0
votes
1answer
1k views

Calculating the average force (mathematical physics)

I am trying to calculate the average force with regards to this question: Calculate the average force that USA gold medalist Allyson Felix (mass = $55.3$ kg, height = $168$ cm) exerted backwards on ...
1
vote
4answers
92 views

Convert $16x^2+56x-80=0$ to the form of $(\text{something})^2=D$

Sorry about this very basic question, I want to convert the equation $16x^2+56x-80=0$ to the form $(\text{something})^2=D$, I know that the answer is $(4x+7)^2-129$, but how can I convert this ...
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votes
2answers
72 views

proving $\frac{1}{n+3}+\frac{1}{n+4}+…+\frac{1}{2n+4}>\frac{1}{2}$

how can one prove that: $\frac{1}{n+3}+\frac{1}{n+4}+...+\frac{1}{2n+4}>\frac{1}{2}$ For all natural $n$, without using induction? thank you.
3
votes
4answers
116 views

How to calculate $9^{47^{51}} \mod 67$?

I've looked at some other related things on here, but this seems a little more complicated with the double exponentiation. Is there a general algorithm to calculate $a^{c_1^{c_2^{...^{c_n}}}} \mod p$ ...
3
votes
5answers
158 views

Why does basic algebra provide one value for $x$ when there should be two?

I have the equation $x^2=x$. If I divide $x$ from both sides I get $x=1$. Yet clearly $x$ can also equal $0$. What step in this process is wrong? It seems to me that there's only one step. And isn'...
-2
votes
2answers
45 views

Real Analysis (Proof)

I'm thinking that maybe this is an application of the Mean Value Theorem. But I'm not sure how to do it. Please help. >.< i) Let $a>0$ and $n>2$. If $$\frac{a}{1+2a}< \frac{1}{n}$$ , ...
2
votes
1answer
39 views

Prove that $f(x+z)$ has $4$ roots $\pm \alpha$ and $\pm \beta$

Let $a$ be a real parameter such that $$f_a(x)= x^4-6x^3+11ax^2-3(2a^2+3a-3)x+1$$ has has four distinct complex roots, that form a parallelogram when plotted on the Argand diagram. Prove That $...
0
votes
1answer
49 views

Exponential Quadratic Equation $-3\left(\frac{2}{3}\right)^x + 2 = x^2-2$ [closed]

How do you solve this equation? $-3\left(\frac{2}{3}\right)^x + 2 = x^2-2$. I have no clue how to do this and any help would be appreciated.
0
votes
1answer
24 views

Solve Graphically

Solve the given systems of equations by graphical method: $$x^2+y^2=5$$ and $$y=2x$$ My Attempt Let's have a look at the second equation ; $$y=2x$$ This is a linear equation in two variables ...
0
votes
0answers
55 views

How do I rearrange this equation without *any* form of $x$ on the other side?

It's easy to solve for $x$ in this equation: $$y=x-f(x)-1$$ where $f(x)$ is a different function than y, but I need to solve for $x$ without $f(x)$ on the other side. How would I accomplish this? ...