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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2answers
21 views

Solution to simple algebra problem

I don't seem to be able to solve this for $x$: $$y = \frac{e^x + e^{-x}}{2}.$$ Ans. is $$x = \ln\left(y\pm\sqrt{y^2-1}\right),$$ but I'd appreciate seeing the intermediate steps. Thanks.
1
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2answers
28 views

Find the locus of points whose distances from the line $y=\sqrt3x$ and x-axis are equal.

Find the locus of points whose distances from the line$\hspace{0.2cm}$ $y=\sqrt3x$$\hspace{0.2cm}$ and x-axis are equal. My solution:I start with the following ...
0
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2answers
47 views

Seeking proof using mathematical induction

\begin{equation}a: \mathbb N ×\mathbb N \to \mathbb R \end{equation} where for all \begin{equation}x,y\in\mathbb N\end{equation}\begin{equation}a(x,y) =a(y,x)\end{equation} How do I show that the ...
-1
votes
1answer
20 views

Find the intersection between a linear equation and a quadratic equation, where the linear equation is x, not y

What the title says essentially. Where one is x=0.6(y-4)^2+4 and another is x=15 The confusing part for me is that it is x=15, there is no y. There are a lot of help for y=?, but x=? NEVERMIND, I ...
0
votes
3answers
41 views

Proving a function has real roots

I am not interested in finding roots but interested in proving that the function has real roots. Suppose a function $f(x) = x^2 - 1$ This function obviously has real roots. $x = {-1, 1}$ How could ...
2
votes
1answer
34 views

Is it possible to find integer solutions for this equation?

I just thought of something really crazy off the top of my head, $$(2+3^{m/n})^{n/m}=(3+2^{n/m})^{m/n}.$$
1
vote
1answer
27 views

simplifying expression of roots of cubic equation

I came across with this question about roots of polynomials. Suppose $a$, $b$ and $c$ are the roots of $x^3-4x+1=0$. Find the value of $ \frac{a^2bc}{a^3+1}+\frac{ab^2c}{b^3+1}+\frac{abc^2}{c^3+1}. $ ...
0
votes
1answer
19 views

Working with sets and its laws

Given: $(A \oplus B) \cup C = (A \cap C) \oplus ( B - C )$ Work with algebra of sets to prove the proposition above is true. In order to give a solution to this problem I've done the process ...
1
vote
1answer
46 views

Prove or disprove $ p^{r+s} \mid (q^{ke} - 1) \iff p^s \mid k$.

Let $p$ be a (odd) prime and $q$ be a power of prime. Suppose $e := \min\{\, e \in \mathbb{N} \mid p \mid (q^e - 1) \,\}$ exists. Put $r := \nu_p(q^e - 1)$ (that is, $p^r \mid (q^e - 1)$ and $p^{r+1} ...
2
votes
2answers
19 views

The width of a rectangle is 3 less than twice its length. If the area of the rectangle is 131 cm^2, what is the length of the diagonal?

The width of a rectangle is 3 less than twice its length. If the area of the rectangle is 131 cm^2, what is the length of the diagonal? I set up the basic equation to solve for l and I know I need ...
2
votes
2answers
27 views

Can the following equations be solved without the need of numerical methods?

I'm taking advanced algebra in school. I have been asked to solve two equations: $\log_{6}(1-x) + \log(x^{2}-9) = 2 \\$ $ 3^{x+2} + 2^x = 5 $ The teacher said this equations can be solved ...
-4
votes
0answers
22 views

ALGEBRA: 2x+3-7k=Z, solve for Z if x,k are units . please help. [on hold]

do not understand at all , please help. intro algebra I have tried many things such as solving normally but still need help please its an exam question (practice)
1
vote
3answers
281 views

Solve the equation

$$9^x=5^x+2\sqrt{20^x}+4^x$$ I'm not really sure where to start. I tried simplifying with logarithms and factoring out the x but it ended up looking just as complicated..
0
votes
0answers
20 views

Binomial Theorem: altrnate ways of writing the coefficent

Is it wrong to write the general term for $(a+b)^n = C(n,n)a^{n} + C(n, n-1) a^{(n-1)}b + C(n, n-2) a^{(n-2)} b^2 + ...+C(n,0)b^n$
1
vote
4answers
83 views

Not understanding how to factor a polynomial completely

$$P(x)=16x^4-81$$ I know that this factors out as: $$P(x)=16(x-\frac { 3 }{ 2 } )^4$$ What I don't understand is the four different zeros of the polynomial...I see one zero which is $\frac { 3 }{ 2 ...
0
votes
3answers
34 views

Under what conditions will one solution of $ax^2+bx+c = 0$ be the reciprocal of the other?

Under what conditions will one solution of $ax^2+bx+c = 0$ be the reciprocal of the other?
0
votes
2answers
26 views

Stuck finding the zeros of a polynomial (complex and real)

Stuck finding the zeros of this polynomial (complex and real): $$x^4+2x^2+1$$ I am not sure how I would factor this. The constant value is really throwing me off. I just need a hint on how to get ...
0
votes
1answer
9 views

Transformations of function not vertical/horizontal

I do not understand how to solve these transformation of functions. Can someone please explain how these are solved? http://i.imgur.com/uWLhxJP.png and http://i.imgur.com/HeytcX2.png Thank you
0
votes
2answers
38 views

$64 = a^2-b^2$, how many solutions are there

$64$ was written as a sum of two squares of two natural numbers. How many solutions does this equation have? $$64 = x^2-y^2 = (x+y)(x-y)$$ For example this works for $10$ and $6$, but how can I ...
0
votes
1answer
30 views

System of equations with parameter

I have been trying to solve this problem for a week now. It goes like this: Find all values of $a$ for which the system $$ \begin{cases} x^2-2x+y^2 = 1 \\[1ex] \dfrac{x+|x|}{y-a}=2 \end{cases} $$ has ...
0
votes
2answers
27 views

Complex power of a real number

What is the meaning of $(-1)^{i}$, where $i^{2}=-1$ and what is its value?
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3answers
55 views

How do I show algebraically that the period of the tangent function is $\pi$?

How do I show that the positive real number $p$ for which $\tan (x+p)=\tan (x)$ is equal to $\pi$? In essence how do I prove the period of the tangent function is $\pi$? Please bear in mind I am a ...
1
vote
1answer
21 views

Integer solutions for equation with two variables

Can someone explain how wolfram alpha calculates integer solutions for these kind of equations: $$ m=\frac{681+13973k}{2021} $$ and how can I do this myself on the paper? Here is a link to the ...
0
votes
1answer
25 views

Polynomial function theorem

The following is from Spivak chapter 3 page 49: Prove that for any polynomial function $f$, and any number $a$, there is a polynomial function $g$, and a number $b$, such that $f(x)=(x-a)g(x)+b$ for ...
0
votes
3answers
60 views

Rearranging the polynomial $x^3-23x^2+142x-120$ prior to factoring it

In the example 15: They are saying that, $$x^3-23x^2+142x-120 = x^3-x^2-22x^2+22x+120x-120$$ From where did $22x^2$ and $22x$ come and also $120x$. Please help me clear my confusion.
0
votes
1answer
30 views

Calculating a floor sum

Is there any explicit closed form expression for $\sum_{k=1}^{\dfrac{p-1}2} \bigg\lfloor \dfrac{kq}p \bigg\rfloor-\bigg\lfloor \dfrac{k(q-1)}{(p-1)} \bigg\rfloor$ , where $p,q$ are odd primes ?
0
votes
3answers
23 views

How to determine different absolute value equation cases?

This is a question from this post. From: $$ |3x|=\left\{ \begin{align} 3x & \text{ , if }x\geq 0 \\ -3x & \text{ , if }x <0 \end{align} \right\} $$ $$ |4x+1|=\left\{ \begin{align} ...
0
votes
2answers
36 views

What are the tangents and asymptotes to $(x-1)(x+1)(x-3)$?

What are the tangents and asymptotes to $(x-1)(x+1)(x-3)$? The equation $$\frac{dy}{dx}=0$$ is not solvable so there are no tangents parallel to x-axis. The function is increasing and it has no ...
2
votes
3answers
34 views

Exchange of volume between spheres until they no longer intersect

Let's say we have two spheres (named $1$ & $2$) whose sizes are stored as radii ($r_1$ and $r_2$) with centres being $d$ distance apart. Now when the two spheres intersect (id est, $d < r_1 + ...
-4
votes
0answers
47 views

Identifying Symbols [on hold]

When you see $x$ written on a piece of paper you automatically identify it. When you yourself write $x^2 + 2x = 0$ The $x$ you write in $x^2$ differs from the $x$ you write in $2x$ just by a ...
3
votes
3answers
28 views

Finding all solutions of an expression and expressing them in the form $a+bi$

$$6x^2+12x+7=0$$ Steps I took: $$\frac { -12\pm \sqrt { 12^{ 2 }-4\cdot6\cdot7 } }{ 12 } $$ $$\frac { -12\pm \sqrt { -24 } }{ 12 } $$ $$\frac { -12\pm i\sqrt { 24 } }{ 12 } $$ $$\frac { -12\pm ...
1
vote
1answer
28 views

Prove this by the principle of mathematical induction.

If $S_r(n)=1^r+2^r+\cdots+n^r$, then prove that $S_r(n) \geq \int_0^nx^r\,dx$. Please help me to solve this problem. I am not able to prove that $P(k+1)$ is true using $P(k)$.
1
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0answers
44 views

Pre calculus Unit Circle

Suppose that you did not have the Unit Circle on Circle A, but rather a circle of radius $5$. Will the angle measures in degrees and/or radians change? Why or why not? Suppose that you did not have ...
3
votes
1answer
19 views

proof regarding the commutativity of an arbitrary oddball binary operator?

My niece has shown me a problem for her advanced high school algebra class that I am personally finding fascinating, regarding the proof (or lack thereof) of the commutativity of a particular ...
1
vote
1answer
19 views

The power output of an amplifier is 6W. The power gain is 80. What is the input power?

The power output of an amplifier is 6W. The power gain is 80. What is the input power? so we know that $$A_p=80$$ $$V\text{ out}= 6 \text W$$ $$V\text{ input} =?$$ Power Gain is $80$ $$\text{ ...
1
vote
1answer
27 views

Find the least integral value of $t$ for which the roots of equation $x^2 + 2(t+1)x + 9t -5=0$ are unequal negative numbers.

My attempt at the answer starts with $(x+p)(x+q)$. And then I got $pq=9t-5$ and $p+q= 2t-2$. Immediately I thought the smallest value for $t$ could be $5$. But when I plug that value into the second ...
0
votes
4answers
61 views

How to simplify $(\sin\theta-\cos\theta)^2+(\sin\theta+\cos\theta)^2$?

Simplify: $(\sin \theta − \cos \theta)^2 + (\sin \theta + \cos \theta)^2$ Answer choices: 1 2 $ \sin^2 \theta$ $ \cos^2 \theta$ I am lost on how to do this. Help would be much appreciated.
1
vote
1answer
41 views

Exercise in algebra - Express different terms

We have $ a = \dfrac{1}{\sqrt{1-b^2}}, c = \sqrt{\dfrac{1+b}{1-b}}, 0 \leq b < 1 $ Express $b$ in terms of $a$, $b$ in terms of $c$, $c$ in terms of $a$ and $a$ in terms of $c$. So I ...
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votes
0answers
29 views

Interesting property of function $\min(a,b).$ [on hold]

Let $a,b$ be real numbers. Prove that $$ \min \left( 0,\min \left( 0,\min \left( 0,\min \left( 0,b \right) -a \right) -b \right) -\min \left( 0,b \right) +a \right) -\min \left( 0 ,\min \left( 0,b ...
0
votes
1answer
33 views

Know any “real life” optimization problems? (Constructing Functions)

Does anyone know "real world" optimization problems? The ones that relate to maximizing area and volume seem a bit contrived. For example, remember this old problem? An orchard has 800 orange ...
0
votes
3answers
76 views

If $x^2 - 3x + 1 = 0$, what is the value of $x^2 + (1/x)^2$?

If $x^2 - 3x + 1 = 0$, what is the value of $x^2 + \left(1/x\right)^2$ ? A. $7$ B. $\frac{7 − 3\sqrt 5}2$ C. $9$ D. $\frac{7 + 3\sqrt 5}2$ E. $10$ I don't really know how to solve this. I tried ...
1
vote
0answers
15 views

What assumptions should be made?

take a problem like A trough is 12 feet long and 3 feet across. Its ends are isosceles triangles with altitudes of 3 feet. Water is being pumped into the trough at 2 cubic feet per minute. How fast ...
0
votes
1answer
14 views

a factor in the numerator is the opposite of the denominator - simplifies to -1

I'm working on a little khan academy problem, finding the limit as x -> 36 in the solution the program explains in the last step that since there are opposite ...
5
votes
7answers
549 views

What is the limit of this specific function?

Please evaluate the following limit for me: $$\lim_{x \to -1} \frac{\sqrt{x^2+8}-3}{x+1} $$ I'd tried my best to solve this but unfortunately, it's too difficult for me. I tried multiplying by its ...
2
votes
1answer
158 views

Simplifying big expression

What to do with this? $$f(x) = \frac{\sinh(\pi)}{\pi} + \frac{2\sinh(\pi)}{\pi}\sum_{n=1}^\infty (-1)^n \left[\frac{\cos(nx)-n \sin(nx)}{1 + n^2}\right]$$ Can it be simplified?
0
votes
2answers
31 views

Why is that for any trigonometric function $f, f(2\pi + \theta )=f(\theta )$ for any value of $\theta$ [proof reading]

Here was the question asked to me :: Why is that for any trigonometric function $f, f(2\pi + \theta )=f(\theta )$ for any value of $\theta$ I spontaneously said that it was because of their very ...
0
votes
1answer
21 views

Where do I go wrong? (cartesian form of parametric eq.)

Basic stuff but I'm not sure where I'm going wrong. Much appreciated if someone could check through my working to see where I misstep! $$x(t)=t+\frac{1}{t}, y(t)=1-\frac{1}{t}$$ So: ...
0
votes
2answers
42 views

Solve Identity about Combination

Find the values of a and b such that $\binom{2n}{2} = a\binom{n}{2} + b(n^2)$ This is a past year question about Introduction of Combinatorics in my university.
0
votes
0answers
21 views

What is the graph of the identity function in cartesian coordinates?

Why is that the graph of f(x) = x is the straight line that is the bisector of the first quadrant? (or, amounting to the same thing, the bisector of the third quadrant) By calculating the outputs for ...
0
votes
1answer
23 views

non linear simultaneous equation with exponentials

I'm having a difficult time finding the coefficients of these set of equations: $A_1 + A_2 + A_3 + A_4 = 0 $ $A_1 + A_2 - A_3 - A_4 = 0 $ $A_1e^{\pi/2} + A_2e^{-\pi/2} - A_3e^{j\pi/2} - ...