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
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1answer
175 views

How many games must a team win to raise its winning percentage from $50\%$ to $60\%$ after playing $60$ games?

A baseball team has won 50% of the 60 games it has played. Find the number of games the team must win in succession to increase it's winning percentage to 60% Please show all work.
0
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0answers
54 views

Let $r$, $s$, $t$ be the roots of the equation $x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$? [duplicate]

the question is mentioned in my math olympiad. please explain how to solve the problem. I have factorised the equation to $-x^2+1$, $x-6x$, $-5x +1$. I am only in year 6.Help!
4
votes
5answers
447 views

Let $r,s,t$ be the roots of the equation $ x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$?

Let $r,s,t$ be the roots of the equation $ x^3 - 6x^2 + 5x + 1$. What is the value of $(2-r)(2-s)(2-t)$? The question is mentioned in my math olympiad. Please explain how to solve the problem. I have ...
3
votes
3answers
905 views

Show that $\tan {\pi \over 8} = \sqrt 2 - 1$

Show that $\tan {\pi \over 8} = \sqrt 2 - 1$, using the identity $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan }^2}\theta }}$ Using $\tan 2\theta = {{2\tan \theta } \over {1 - {{\tan ...
0
votes
3answers
976 views

Prove algebraically that $(\sqrt{3}+1)/(\sqrt{3}-1)=2+\sqrt{3}$

This is indeed an identity, but I cannot seem to understand how you would go about proving these two equations are equal algebraically? Do you see a method to prove it? ...
1
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2answers
95 views

If $a,b$, and $c$ are reals satisfying $ \frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}=6$, calculate $ x = \frac{(a+b+c)^3}{a^3+b^3+abc}$

$$ \frac{b+c}{a}+\frac{a+c}{b}+\frac{a+b}{c}=6$$ $$ x = \frac{(a+b+c)^3}{a^3+b^3+abc}$$ As a trivial solution, I found $$a=b=c$$ then, $x$ will always be $9$. Despite this, my algebra teacher told me ...
2
votes
1answer
91 views

Partial Fraction Decomposition

I am really stuck on how to do this. It is a step I need to do for an inductive proof. I have $$\frac{1}{n(n-1)} $$ Do I set it up like this: $\frac{ A}{n} + \frac{B}{n-1}$ ? $$1= A(n-1) + ...
1
vote
4answers
117 views

How many intersections has the following two curves at the point $(0,0)$?

The curves are as follows: $y^2+yx^2-x^3$ and $y^2-x^5$
8
votes
9answers
960 views

If $x \neq 0,y \neq 0,$ then $x^2+xy+y^2$ is …

I came across the following problem that says: If $x \neq 0,y \neq 0,$ then $x^2+xy+y^2$ is 1.Always positive 2.Always negative 3.zero 4.Sometimes positive and sometimes negative. ...
1
vote
1answer
80 views

Finding the equation of the plane which consists two lines

The problem: Given $$L_1: x= -1+4t, y=3+t, z=1$$ $$L_2:x=-13+12t, y=1+6t, z=2+3t$$ Find the plane which consists $L_1$ and $L_2$. So I found that $L_1$ and $L_2$ intersect at $(-17, -1, 1)$. But ...
0
votes
1answer
333 views

Angle between 2 faces of a pyramid

The problem: Given a pyramid with $P_0=(0,0,0)$, $P_1=(1,1,1)$, $P_2=(2,-1,2)$, $P_3=(3,0,1)$, find the angle between the $P_1P_2P_3$ face the $P_0P_1P_2$ face. My idea for the solution is to ...
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votes
2answers
4k views

Finding the range of function - square root

find the range of y = $\sqrt{x-1}+\sqrt{5-x}$ Since domain of square root function is defined for $f(x) \geq 0$ therefore : $\sqrt{x-1} \geq 0 ; x \geq 1 $ also $\sqrt{5-x } \geq 0 ;\: x \leq 5$ ...
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2answers
83 views

Larger Theory for root formula

Consider the quadratic equation: $$ax^2 + bx + c = 0$$ and the linear equation: $$bx + c = 0$$. We note the solution of the linear equation is $$x = -\frac{c}{b}.$$ We note the solution of the ...
1
vote
3answers
159 views

How do I prove that $2\cos (2\theta + {\pi \over 3}) \equiv - 2\sin(2\theta - {\pi \over 6})$

Using the identity $\cos (\theta + {\pi \over 2}) \equiv - \sin\theta $
2
votes
1answer
28 views

Constructing a specific polynomial?

I need to construct a polynomial with zeroes at $3$, $5$, and $10$ (and the function can't just be tangent at those points, it has to go below/above the axis. Also, there can't be any zeroes in ...
0
votes
2answers
56 views

Basic algebraic equation question

Ok thats very basic but dont get it , I have the following equation $R-A X-B \dfrac{A X}{2 B}=0$ the right result of it is $R- \dfrac{3 A X}{2}=0$ well, now when I attempt to solve it I do the ...
3
votes
4answers
126 views

Solve $\frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+\cdots\right)$

If $$\displaystyle \frac{1}{2x}+\frac{1}{2}\left(\frac{1}{2x}+ \frac{1}{2}\left(\frac{1}{2x} +\cdots\right) \right) = y$$ then what is $x$? I was thinking of expanding the brackets and trying to ...
5
votes
6answers
3k views

How do I prove: $\cos (\theta + 90^\circ) \equiv - \sin \theta $

How do I go about proving this? I know one method is: $\eqalign{ \cos (90^\circ + \theta ) &\equiv \cos90^\circ \cos\theta - \sin90^\circ \sin\theta \cr & \equiv (0)(\cos\theta ) - ...
4
votes
4answers
10k views

What are the rules for factorial manipulation?

I know that $$(k+1)! - 1 + (k+1)(k+1)! = (k+2)! - 1$$ thanks to wolframalpha, but I don't understand the steps for simplification, and I can't seem to find any rules about factorial manipulations ...
1
vote
1answer
66 views

How to factor $2b^2c^2 + 2c^2a^2 + 2a^2b^2 -a^4-b^4-c^2$?

The term is: $2b^2c^2 + 2c^2a^2 + 2a^2b^2 -a^4-b^4-c^2$ And the answer is : $(a+b+c)(b+c-a)(c+a-b)(a+b-c)$ I have tried a lot, but could't accomplish. Please don't bring up any complex method, it ...
0
votes
0answers
24 views

Relation of two functions defined via real-valued Möbius transfrom

Consider $u(x) := (a x + b)/(c x + d)$ where $a,b,c,d > 0$ and we take $x \in (0,\infty)$. Let $v(x) := u(1/x)$. Consider the two functions $$ F(x,y) := \frac{1+u(x)}{1+v(y)},\quad G(x,y) = ...
1
vote
4answers
711 views

Find the value of $\alpha $ given $2\sin\theta -\sqrt 5 \cos \theta \equiv - 3\cos (\theta + \alpha )$

Given: $$2\sin\theta -\sqrt 5 \cos \theta \equiv - 3\cos (\theta + \alpha ),$$ where $$0 <\alpha < 90^\circ, $$ find $α.$ The issue I have with this question is the $-3$ on the right hand ...
0
votes
1answer
218 views

Calculating the unknown value by using available slope value

I have the temperature value for one hour with 2 minutes frequency. In other words 30 measurements for 1 hour. I have calculate the slope between first and last measurements. Consider a situation ...
3
votes
0answers
52 views

Need $f:[0,\infty)\to[0,\infty)$ such that $f$ is not convex but $f(x^p)$ is for $p>1$.

To be more specific I need to find an $f:[0,+\infty)\to[0,+\infty)$ which satisfies the following: (somewhat trivial stuff) The function $f$ is continuous, nondecreasing, there exists $k>0$ such ...
4
votes
1answer
137 views

Trigonometry - Calculating the pyramid volume

The problem: There be the points $P_0(0,0,0)$, $P_1(1,1,1)$, $P_2(2,-1,2)$ and $P_3(3,0,1)$. Calculate the volume of the pyramid. Now I assumed the base of the pyramid is a triangle, with points ...
1
vote
1answer
60 views

Trouble with rewriting formulas in different forms

We have the following formulas: $$ a = \dfrac{\displaystyle\sum (x-\bar{x})(y-\bar{y})}{\displaystyle\sum(x-\bar{x})^2}$$ $$ r = \dfrac{ \displaystyle \sum ...
3
votes
3answers
63 views

Triplets 4th of power of first equals to sum of other two

I was looking through the admission test for the University Normale of Pisa and I found a problem that I don't know how to solve, it state something like that: Find all the triplets of number (x, y, ...
0
votes
1answer
102 views

Calculate the time after which Money doubles

I while solving a problem in banking just thought to form a formula for the time period after which money deposited in bank at a compounded interest rate @$\alpha $ % p.a. . Amount for compounded ...
5
votes
2answers
308 views

$ \sin^{2000}{x}+\cos^{2000}{x} =1$ equation explanation

Solve the equation: $$ \sin^{2000}{x}+\cos^{2000}{x} =1.$$ What I did: $\sin^2{x} =1 \land \cos^2{x}=0$ when $x=\frac \pi2 + \pi k $ $\cos^2 {x} =1 \land \sin^2{x}=0$ when $x= \pi k$ I think ...
1
vote
1answer
33 views

Adjust percentage

I'm stuck on what I would think is a simple problem. A group of $3$ people are selling a product through a store. Under the current arraignment, the store gets $30$% of the price the product is sold ...
1
vote
1answer
98 views

Convergent & Cauchy Sequence related prove

(1) Consider the two convergent sequences $\{a_n\}$and $\{b_n\}$ such that $\{a_n\}\to a$ and $\{b_n\}\to b$. Prove that $\{a_n+b_n\}\to a + b$ (2) Prove that a convergent sequence is Cauchy. ...
-1
votes
1answer
126 views

Quadratic Baseball Question

The height of a baseball is modeled by the function $h(x)=-0.005x^2+0.3x+1.5$, would an outfielder which is modeled by the function $m(x)=-0.06x+5.6$ where $50 \le x \le 90$, catch the ball?
0
votes
2answers
412 views

Intersection between a line and a wave

Is it possible to find the points of intersection between a line and a sine wave? I would like a function to find the nth intersection, rather than just first intersection within the domain of the ...
3
votes
2answers
2k views

What does it mean to eliminate $\theta$ from these equations? How should I do it?

Eliminate $\theta$ from the following pairs of equations: A) $x=\sin \theta$, $y=\sin 2\theta$ B) $x=3\cos 2\theta +1$, $y=2\sin \theta$ My problem is I really don't understand what ...
5
votes
6answers
274 views

Formula for the $1\cdot 2 + 2\cdot 3 + 3\cdot 4+\ldots + n\cdot (n+1)$ sum

Is there a formula for the following sum? $S_n = 1\cdot2 + 2\cdot 3 + 3\cdot 4 + 4\cdot 5 +\ldots + n\cdot (n+1)$
1
vote
1answer
130 views

Alternative solutions to $n^2+n = k^2+k + 2kn$

Consider this equation: $n^2+n = k^2+k + 2kn$ I want to find the set of non-negative integer n,k that satisfies the equation. I tried to write $n$ as $k$ by solving the equation with $n$ as root ...
1
vote
1answer
195 views

Functions - proving injectivity or surjectivity

Consider a function h: $\Bbb Z \times \Bbb Z \rightarrow \Bbb Q$ defined as $\displaystyle h(m,n) = \frac{m}{|n|+1}$ Determine whether this is injective and whether it is surjective. Since this ...
3
votes
2answers
339 views

Proving $\tan (\frac{\pi }{4} - x) = \frac{{1 - \sin 2x}}{{\cos 2x}}$

How do I prove the identity: $$\tan \left(\frac{\pi }{4} - x\right) = \frac{{1 - \sin 2x}}{{\cos 2x}}$$ Any common strategies on solving other identities would also be appreciated. I chose to ...
3
votes
2answers
53 views

Calculations with Percentages

A group has 60% female and 40% male members. 42% of all group members are older then 60 years, among men even 60% are older then 60. a) which percentage of the women is younger then 60? b) If a young ...
6
votes
1answer
89 views

Find the floor value of a finite continued surd

Given $x=20062007$, and let $$A=\sqrt{x^2+\sqrt{4x^2+\sqrt{16x^2+\sqrt{100x^2+39x+\sqrt{3}}}}}.$$ Find the greatest integer not exceeding $A$.
0
votes
1answer
60 views

A question about roots for a quartic equation

Let $k\in\mathbb{R}$ and $f(x)=x^{4}+(2-k^{2})x^{2}-2k^{2}x+(1-k^{2})$. Suppose that there are exactly two distinct real roots for the equation $f(x)=0$. Find the range of $k$. I don't know the ...
0
votes
2answers
61 views

Finding a function from the value of an inverse fuction

The function $f(x) = k(2 - x - x^3)$ has an inverse, and $f^{-1}(3) = -2$. Find $k$. I tried setting $f(x)$ equal to $3$ and plugging $-2$ into $x$ and I ended up with $3 = k(12)$. I'm not sure ...
5
votes
2answers
927 views

How can I make sure I never forget.

I am currently refreshing my Elementary Algebra using Schaum's Outlines. I find them useful as they are choc full of exercises (Which I now realize is the only way to master algebra). I am worried ...
6
votes
2answers
108 views

Solve $\lceil x\rceil^2+\lceil x+1\rceil^2=25$

How can we find the solution set of $\lceil x\rceil^2+\lceil x+1\rceil^2=25$ where $\lceil x\rceil$ is the ceiling function?
2
votes
2answers
85 views

How to find the area/volume of a can?

I'm doing some area work and I was wondering how find the area of a can... It's like finding the area of a cylinder right? I'm really confused. I just need the formulae... thanks
1
vote
2answers
171 views

Easy way to simplify this expression?

I'm teaching myself algebra 2 and I'm at this expression (I'm trying to find the roots): $$ x=\frac{-1-2\sqrt{5}\pm\sqrt{21-4\sqrt{5}}}{4} $$ My calculator gives $ -\sqrt{5} $ and $ -\frac12 $ and ...
3
votes
2answers
191 views

$2x^4 \le\sin^4x+\cos^6x -1 $ inequality

Solve the inequality: $2x^4 \le \sin^4x+\cos^6x -1 $ What I did: Using trig. identities I simplified the RHS to: $3\sin^2(x)\cos^2(x) + (\sin^2(x))\cos^2(x)\sin^2(x)$ but still I don't know where to ...
4
votes
1answer
84 views

How many natural numbers $n$ exist for $n=a^2−b^2−c^2$

How many natural numbers $n≤1000$ cannot be written in the form $a^2−b^2−c^2 \ ;$ where $a$,$b$ and $c$ are non-negative integers subject to condition $a≥b+c$. How to approach?
0
votes
3answers
43 views

The absolute maximum of a function

Let $f(x)=\dfrac{\sqrt{4+32x^{2}+x^{4}}-\sqrt{4+x^{4}}}{x}$, where $x\in \mathbb{R}$ and $x\neq 0$. Suppose that $f(x_{0})=M$ is the absolute maximum of $f$. Find $(x_{0},M)$. I have no good ideas. ...
12
votes
4answers
617 views

Simplifying $\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt {5 +\cdots}}}}$

How to simplify the expression: $$\sqrt{5+2\sqrt{5+2\sqrt{5+2\sqrt{\cdots}}}}.$$ If I could at least know what kind of reference there is that would explain these type of expressions that would be ...