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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Simplifying calculation

My education system won't allow me to use calculator even though within complex number. Luckily, we use multiple choice (which I can do approximation) I am no human calculator, and if I count, it ...
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1answer
53 views

Calculation of total number of real ordered pairs $(x,y)$

Calculation of total number of real ordered pairs $(x,y)$ in $x^2-4x+2=\sin^2 y$ and $x^2+y^2\leq 3$ $\bf{My\; Try::}$ Given $x^2-4x+2=x^2-4x+4-2=\sin^2 y\Rightarrow (x-2)^2-2=\sin^2 y$ ...
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4answers
56 views

$a+b=3$ and $a^2+ b^2=7$. Find $a\cdot b$

So we have $a+b=3$ and $a^2+ b^2=7$. I have to find $a\cdot b$.
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4answers
79 views

How to find the domain of this function?

f(x)= $\frac{(\sqrt{x}-\sqrt{x-1} )}{( \sqrt{x}+\sqrt{x-1} )}\;$ first off $\sqrt{x}$ is defined for: $$x > 0 \tag{1}$$ and $\sqrt{x-1}$ is defined for: $$x \ge 1 ...
4
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3answers
55 views

Simplifying the result formula for depressed Cubic

After understanding the Cardano's formula for solving the depressed cubic (of the form $x^3+mx=n$, of course), I tried to find the solution of the equation $$x^3+6x=20.$$ After plugging into the ...
4
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2answers
60 views

Convert from Nested Square Roots to Sum of Square Roots

I am looking for a way to easily discover how to go from a nested root to a sum of roots. for example, $$\sqrt{10-2\sqrt{21}}=\sqrt{3}-\sqrt{7}$$ I know that if i set ...
3
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0answers
31 views

Selling oranges when people queue up in a line

We have $a$ oranges to give to $b$ people. Each person has a value $f(n)$ for receiving $n$ oranges, where $f$ is a nondecreasing, nonnegative function that is the same for everyone. Let $X$ be the ...
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1answer
46 views

Solve $(x^2+1)^2=4x(1-x^2)$

Let $(x^2+1)^2=4x(1-x^2)$. I haven't tried anything real yet, except to expand. I know it is easy but I don't have any idea for the moment. So please help!
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2answers
61 views

How to solve this algebra expression?

I'm attempting work out why : $$\frac{(6x/x) - (9/x)}{(x/x) - (1/x)}=6$$ So far I have : $$\frac{(6x/x) - (9/x)}{(x/x) - (1/x)} =$$ $$\frac{(6x/x) - (9/x)}{(1) - (1/x)} =$$ Specifically how to ...
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0answers
35 views

How much information is missing?

If we know the value of $\frac{(a-b)}{(c-d)}$, can we calculate the value of $\frac{(a-d)}{(c-b)}$ That is : Let $\frac{(a-b)}{(c-d)}=k$ , can we calculate $\frac{(a-d)}{(c-b)}$ in terms of $k$ And if ...
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1answer
30 views

Must a rational function always be in numerator-denominator form?

This may seem trivial, but I'm looking at two examples from high school math books and wondering if they are really examples of rational functions. The first is a line $\overline{DT}$ made up of two ...
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6answers
62 views

Solve for $x$ : $\log_e(x^2-16)\lt \log_e(4x-11)$

$\log_e(x^2-16)\lt \log_e(4x-11)$ My attempt: Since the base is $\gt 1$, we have from the above , $$x^2-16-4x+11\lt 0\\ \implies x^2-4x-5\lt 0\\ \implies(x-5)\cdot (x+1)\lt 0$$ If I say $(x-5)\gt ...
1
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0answers
45 views

To find root of $x^n+1=0$ [duplicate]

If $ \alpha_1,\alpha_n, ....\alpha_n $ be the roots of the equation $x^n+1=0$, then $(1-\alpha_1)(1-\alpha_2)...(1-\alpha_n)$ is equal to a) 1. b) 0 c)n d)2 when I put n=3,and directly evalutae ...
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1answer
31 views

finding root of an equation with real coefficient.

If the equation $x^4 + ax^3 + bx^2 + cx+ 1=0 $ (where a,b,c are real numbers) has no real roots and if at least one root is of modulus one, then a)b=c b)a=c c)a=b d)none of the above
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2answers
32 views

Questions about Exponentiation and roots and logarithms.

in this page a few questions I want to ask you about the Exponentiation and roots and logarithms: What and how the Exponentiation definition can be defined by real numbers.? What is the overall ...
2
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2answers
479 views

Highschool Exam Question About Cube Factoring

Given; $ a^3 - 3ab^2 = 10 $ and $ b^3 - 3ba^2 = 5$ What is the value of $ a^2 + b^2 $ ?
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1answer
37 views

Profit and loss

A fruit seller buys a large quantity of apples for $150\$$ . $200$ of the apples are rotten and he sells each of the remaining apples at $10$ cents more than what he paid and makes a profit of ...
7
votes
3answers
107 views

Find all real numbers $a,b$ such that $|a|+|b|\geq\frac{2}{\sqrt{3}}$ and $|a\sin x+b\sin{2x}|\leq 1$ for all real $x$.

Find all real numbers $a,b$ such that $|a|+|b|\geqslant\frac{2}{\sqrt{3}}$ and $|a\sin(x)+b\sin(2x)|\leqslant 1$ for all real $x$. We could write the inequality as $$ ...
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1answer
53 views

integral problem $\int \frac{2 \lambda a}{\mathbf{ (e^{at}-1)\lambda \sigma^2+2ae^{-at}}}dt $

Does anybody know how to tackle the below integral? I am analyzing a formula derivation where this appears as the final calculation, but I don't know how to get it solved $$\int \frac{2 \lambda ...
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1answer
43 views

Iterative method for finding real solutions to $a+b+c+d = abcd = 7.11$

I have "come up with" a method for finding $a,b,c,d \in \Bbb{R}$ such that their sum and product is equal and wanted to ask if the method is sound. First, rearrange both equations so that only $a, b$ ...
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2answers
46 views

Form all numbers from limited digits

Let $M$ be a subset of the digits $\{0,1,\ldots,9\}$, and $N$ the set of numbers formed with digits in $M$. Suppose that all numbers from $1 $ to $99999999$ either belong to $N$ or can be written as ...
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3answers
87 views

What's the best way to compute $\frac{a^4 + b^4 + c^4}{a^2 + b^2 + c^2}$

So, my teacher gave us this to compute yesterday, and I'm completly confused on how should I proceed : $$\frac{1^4 + 2012^4 +2013^4}{1^2 + 2012^2 + 2013^2}$$ I've tried several ways, but most of ...
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0answers
45 views

Solving $\sin(\theta) + \cos(\theta) = -1$ using the T-Formula [duplicate]

In class we've learned 2 methods for solving trigonometric equations. T-Formula: If you're not familiar with the T-Formula look here; T-Formula Auxiliary Method: If you're not familiar look here ...
2
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2answers
146 views

Solving Three equations for 3 Unknowns

Today I have a question and I am really curious to know about this. Question: $$ 16y+39z+50zy=0$$ $$ 85x-78z+95zx=0$$ $$ 85x+32y+70xy=0$$ $$\text{Are The Equations like these can be solve for ...
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4answers
28 views

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$

Prove using factorials that ${n\choose k}+2{n\choose k+1}+{n\choose k+2}={n+2\choose k+2}$ I think I'm having a bit of algebra problem with this proof. Here is my work thus ...
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1answer
63 views

Mathematical induction proof problem: $\sum_{i=1}^{n-1} i(i+1) = \frac{n(n+1)(n-1)}3$

I am having difficulty proving the inductive hypothesis $(k+1)$ for the following statement: $$\sum_{i=1}^{n-1} (i(i+1)) = \frac{(n)(n+1)(n-1)}{3}$$ This is what I have so far: $$(Step \ 1) ...
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15 views

Questions about simplifying an equation and about finding orthogonal trajectories

First question: a question involves simplifying the following equation (or to be precise, making the equation an explicit function) $$8y^2=x^2(1-x^2).$$ When I tried to simplify it, I got $$y= \pm ...
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0answers
36 views

The formula for arithmetic series doesn't make sense. Why doesn't $na_1 + nd$ work?

In an arithmetic sequence, we can find each term like so: $a_1 = a1$ $a_2 = a_1 + d$ $a_3 = a_1 + 2d$ and so forth. The series then, is $a_1 + a_2 + a_3 + ... + a_n$ Therefore, intuitively, its ...
2
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3answers
87 views

Simple maths - rearranging terms

I have a formula: $$\frac{\sqrt{\frac{2Ka}{h}}}{a}$$ How can it be arranged as: $$\sqrt{\frac{2K}{ah}}$$ I only can do: $$\frac{\sqrt{\frac{2Ka}{h}}}{a}$$ $$=\sqrt{\frac{2Ka}{h}}.\frac{1}{a}$$ ...
2
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1answer
47 views

Why does $\ln x / \ln b = \log_b x$?

I'm doing some Java code. As far as I can tell, Java only has functions that do natural log and base $10$ log. I have a requirement to specify the base. I've seen that doing $\ln x/ \ln b$ is the ...
2
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3answers
47 views

For which values of k will $x^3−3x^2+6x+k=0$ have 3 real roots?

Good day all, I have an example in my book that ask: For which values of k, if any, does $x^3−3x^2+6x+k=0$ have 3 real roots? I know that this is a quadratic equation and that normally I would ...
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1answer
28 views

What is the number of ordered pairs of real numbers (a,b) such that …

Problem : What is the number of ordered pairs of real numbers (a,b) such that $(a+ib)^{2002}=a-bi$ My approach : Multiplying both sides by a+ib we get $(a+ib)^{2003} = a^2+b^2$ $\Rightarrow ...
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0answers
24 views

How to isolate and solve for k in a Sigma notation probability mass function equation?

"isolate and solve for k:" $$P(X = k) = \sum_{k=0}^n {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$ If the above equation is a function of P, how would the equation be stated as a ...
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3answers
43 views

Find a close form expression for $f(x)$

Here is the problem I am currently having trouble with. I have a pretty decent basis on how to do recurrence relations, but the $\frac{1}{n!}$ has got me in a rut. I tried multiplying the right side ...
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2answers
40 views

How to find all Possible values for A and B, given only one equation? [closed]

Given that : $$\frac{1}{a} + \frac{1}{b} = \frac{1}{12} $$ With a & b integers How can I find all possible values of a and b with only one equation (this one ?) . From what I'v learned in ...
4
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1answer
39 views

An inequality $\frac1{(n+1)^{1/(n+1)}}-\frac1{n^{1/n}}\le \frac1{n+1}$

I have graphed the functions $f,g:\mathbb{R^+}\to\mathbb{R}$ defined by $$f(x)=\frac1{(x+1)^{1/(x+1)}}-\frac1{x^{1/x}}\mbox{ and } g(x)=\frac1{x+1}$$ and it seems like $f(x)\le g(x)$ for all $x>0$. ...
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2answers
46 views

Largest solution of $2\ln(e^x-1)=\ln2+\ln(e^x+3)$

I've been trying to solve this equation: $$2\ln(e^x-1)=\ln2+\ln(e^x+3).$$ I am asked for the equation's largest solution, and I come to a point where I get $\ln(e^x(e^x-4)) = \ln5$. I can't find my ...
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2answers
45 views

if $(8a-7b-3c)^2=6(a^2-b^2-c^2)$,find $a:b:c$

Let $a,b,c>0$,and such $$(8a-7b-3c)^2=6(a^2-b^2-c^2)$$ Find $a:b:c$ since $$(8a-7b-3c)^2-6(a^2-b^2-c^2)=58a^2+55b^2+15c^2-112ab-48ac+42bc=0$$? How find $a:b:c=?$
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1answer
22 views

Trouble using Newton's Method on $6$th root of $2$

first post here. I've been reading a calculus text for the last few months and have stumbled into a problem that I cannot find an answer to after all my researching. I am having trouble ...
2
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0answers
47 views

Rewriting the polynomial $x^2 + y^2 + z^2 - xy - xz - yz$?

I am wondering whether it is possible to factorize/rewrite the following with Newton's identities (or some other algorithm) where the polynomial is given by $x^2 + y^2 + z^2 - xy - xz - yz$. I am ...
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1answer
17 views

The number of of integral ordered pairs (x,y) satisfying the system of given equations

Question The number of of integral ordered pairs (x,y) satisfying the system of given equations |x+y-4| = 5 .........(i)and |x-3|+|y-1| = 5 .....(ii) is/are (A)2 (B)4 (C)6 (D)12 My attempt ...
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2answers
49 views

Find all $(x,y)$ such that $1/x+1/y=1/7$.

While working in a workbook, I got stuck on the following problem: Find all the ordered pairs $(x,y)$ such that they satisfy the equation $1/x + 1/y = 1/7$. I made a bit of progress, but got stuck. ...
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2answers
31 views

How can you find two solutions to $\cos(3x - \frac{\pi}{2}) = 0$ by illustrating the situation on the unit circle?

How can you find two solutions to $\cos(3x - \frac{\pi}{2}) = 0$ by illustrating the situation on the unit circle? The solutions I got for this are $x = 0+2\pi k$ or $\pi/3 + 2\pi k$ where $k$ ...
0
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1answer
74 views

If $\alpha^3 - 3\alpha^2 + 5\alpha -17 =0 $ and $\beta^3 - 3\beta^2 + 5\beta+11 =0 $ then find value of $\alpha+\beta$($\alpha,\beta$ is real number) [closed]

The curve $y = x^3 - 3x^2 + 5x $ is a strictly increasing curve. $y=x^3 - 3x^2 + 5x -17 =0 $ intersect x axis between 3 and 4. $y=x^3 - 3x^2 + 5x +11 =0 $ intersect x axis between -2 and -1. The ...
0
votes
4answers
65 views

Question on Time and work problems

A cistern which has a leak on the bottom is filled in $1$ hr. Had there been no leak, it could have been filled in $40$ minutes. If the cistern is full, at what time will the leakage empty it? My ...
0
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0answers
25 views

Find all the solutions of $\sin x=ax$ [duplicate]

I need to find all the solutions to the equation $$\sin x=ax$$ where $a \in \mathbb{R}$. If $\vert a \vert \geq 1$, I know that the only solution is $x=0$. My problem is when $\vert a \vert \leq 1$ ...
3
votes
1answer
75 views

Find real roots of the equation

Find all real solutions to $$\dfrac{\sqrt{x+1}}{2+\sqrt{2-x}} - \dfrac{\sqrt{x^2-x+2}}{2+\sqrt{-x^2+x+1}} = x^3-x^2-x+1$$ This question is very similar to one of my previous problem, ...
2
votes
1answer
47 views

Solving a mixed radical and quadratic equation

Solve for $x \in \mathbb{R}$ $$4x^2(x+2) +3(2x^2-4x-3)\sqrt{4x+3} +6x = 0$$ I tried taking square by isolating the radical, but the resultant equation couldn't be solved. Any help ...
0
votes
1answer
23 views

Deriving formula for asymptotes of a hyperbola

I'm trying to find a precalculus-level derivation of the formula for the asymptotes of a hyperbola. My book says: Solving $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ for $y$, we obtain $y = ...
10
votes
1answer
146 views

Finding all real roots of the equation $(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$

Find all real roots of the equation $$(x+1) \sqrt{x+2} + (x+6)\sqrt{x+7} = x^2+7x+12$$ I tried squaring the equation, but the degree of the equation became too high and unmanageable. I ...