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
15 views

No. f ordered pair $(a,r)$ in Logarithmic equation.

If $a_{1},a_{2},a_{3},.............$ be a Geometric Progression, Where $a_{1} = a$ and common ratio $r$ are positive integers. If $\displaystyle ...
1
vote
3answers
29 views

Why $x=u+v$ substitution works?

I have the solution for the follwoing example : $$x^4+y^4=82$$ $$x-y=2$$ The author substitutes $x=u+v$ and $y=u-v$ My question is: If we have two numbers ($x, y$), can we always find ...
-1
votes
0answers
32 views

Number theory / decimal representation

Prove that for any $n\in\mathbb{N}$ there exists a number $m\in\mathbb{N}$ such that the decimal representation of $m^2$ has $n$ ones at the beginning and some combination of $n$ ones and twos at ...
2
votes
0answers
35 views

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$ [on hold]

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as ...
0
votes
5answers
73 views

Solving $12-\sin(\theta)=\cos(2\theta)$

$12-\sin(\theta)=\cos(2\theta)$ What's the correct answer on the interval $[0, 2 \pi]$. Please help, I'm rather lost. I started with: $$12-\sin(\theta)=1-2\sin^2(\theta)$$ and then I cant get ...
-2
votes
1answer
31 views

How many of each kind? [on hold]

Abby and Bing Woo own a small bakery that specializes in just two kinds of fudge-peanut butter and vanilla. They need to decide how many dozens of each kind of fudge to make for tomorrow. They are ...
2
votes
2answers
39 views

Argument of complex number $(\tan \theta)$

I'm given $-2+2\sqrt{3}i$. The question asks me to find the argument. My attempt, $\tan \theta=\frac{2\sqrt{3}}{2}$ So $\theta=\frac{\pi}{3}$. But the given answer is $\frac{2\pi}{3}$. Why?
-2
votes
0answers
55 views

Does anyone know of a book that explains factoring well?

Can anyone recommend me a book comes well explained the process to factor a polynomial of two variables with complex coefficients, as the multiplication of convergent power series in two variables ...
-1
votes
3answers
62 views

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then $a=b=c=0$ [on hold]

If a quadratic equation $ax^2+bx+c=0$ has more than two roots, then it is an identity i.e. it is true for all values of $x$ and $a=b=c=0$. What is a proof of this?
-3
votes
1answer
48 views

Find the width of a rectangle with an area of $x^2 -4x -12$ and the length of $x-2$

There is a rectangle with an area of $x^2 -4x -12$. The length is $x-2$, what is the width? I'm having serious trouble solving this, can anyone help?
0
votes
3answers
53 views

How to solve the equations of the type $\sin a + \sin b = \sin x$?

I came across a question in my book that's like this: $$\sin20 + \sin40 = \sin x $$ I don't know if the values of the $a$ and $b$ make a difference (or in this case, the fact that $b = 2a$) but I'd ...
0
votes
3answers
69 views

Grade 8 simple algebra equation help

I find this question hard, please help. It is given that $x+\frac{1}{x}=3$ and $x^2+\frac{1}{x^2}=7$. Please find the value of $x^3+\frac{1}{x^3}$. Please show the steps.
0
votes
2answers
111 views

Is this true that $(\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1 \implies A+B+C=\pi)$? [on hold]

Assume that $A,B,C$ are positive real numbers and $A,B,C \in (0,\frac{\pi}{2}]$ and we have $$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C = 1 $$ prove or disprove that $$A+B+C=\pi$$
1
vote
1answer
36 views

Roots of the complex equations

Find all the roots for the following equation. $2x^4-x^3-x^2+3x+1=0$ My attempt, I factorised it to $(x+1)(2x^3-3x^2+2x+1)=0$ So I know one of its roots is -1. How to proceed then?
0
votes
1answer
23 views

Solving an exponential equation by means of factoring

this is my first post here. The equation I could halfway solve is this one: $4^x+4-2^x(2^{x+1}-3)=0$ How do I factor this polynomial? Is there any other way besides factoring?
0
votes
0answers
22 views

Algebra Integral simplification

Let some equation problem final result is like this \begin{align} M=\mathrm{exp}\bigg\{-\pi\lambda v^2+\pi\lambda v^2\bigg(\displaystyle\int_o^s \frac{2x}{v^2}\mathrm{d}x \nonumber \\\\ ...
2
votes
3answers
38 views

A not so hard basic calculus problem? But it appears to be very lengthy

Find the coordinates of the two points on the curve $y=4-x^2$ whose tangents pass through the point $(-1,7)$. My work: Let the two points be $(a,b)$ and $(c,d)$. And $\frac{dy}{dx}=-2x$, so the ...
3
votes
2answers
78 views

How can I prove that $2ab \leq a^2 + b^2$?

I'm stuck with it: $2ab \leq a^2 + b^2$. Have no idea how to go beyond this ($a,b \geq 0$). Thanks!
-1
votes
2answers
20 views

Pricing call options with binomial trees (proof) [on hold]

I need assistance in proving that the following line: $$f = S_0\left(\frac{f_u - f_d}{S_0u - S_0d}\right)\left(1 - ue^{-rT}\right) + f_ue^{-rT}$$ Equals this line: $$f = \frac{f_u\left(1 - ...
0
votes
3answers
36 views

Question on polar coordinates and cartesian coordinates

I know the conversion between polar coordinates and cartesian coordinates. Nevertheless, I cannot understand why $r=2a\cos\theta$ represents a circle of radius $a$ and center $(a,0)$. Can anyone ...
1
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2answers
31 views

Function Composition Thinking Problem

Here is the question: A banquet hall charges $\$975$ to rent a room, plus $\$39.95$ per person. Next month they will offer a $20\%$ discount off the total bill. Determine two equations, one for ...
0
votes
2answers
31 views

Finding two functions $f(x)$ and $g(x)$

I am not sure how to approach this question. It asks to find $f(x)$ and $g(x)$ such that $h(x)=f(g(x))$, for each function: a) $$h(x)=\sqrt{x^2 + 6}$$ b)$$h(x)=\frac{1}{x^3}-7x+2$$ If someone ...
0
votes
1answer
22 views

Determine the value of combined functions with square roots

The question I have is to determine the value of $f(g(x))$ given $f(x)=\sqrt{16-x^2}$ and $g(x)=x^2$ I know generally how to tackle these kinds of questions, but I am not sure what to do when there ...
0
votes
3answers
25 views

Find the domain of combined functions

I have a question asking to find the domain of $g(f(x))$ given $f(x)=2x^2+x$, and $g(x)=x^2+1$. I can easily do these questions in reverse when you have to find $f(g(x))$, but when having to find ...
1
vote
0answers
37 views

Solve $x=C \log(C \log(x+A)+B)$

Is it possible to resolve an equation of the type $$x=C\log{(C\log{(x+A)}+B)}$$ (where $A$, $B$, and $C$ are real-valued parameters) for $x$? As far as I can see, the function on the right hand ...
0
votes
0answers
20 views

free tool for algebraic manipulations of commutator expressions

Is there an (ideally) free tool for algebraic manipulations of commutator expressions of the form: given $$c(A,B):=\tfrac{1}{2}[A,B]+\tfrac{1}{12}[A,[A,B]]+\tfrac{1}{12}[B,[B,A]]$$ simplify (express ...
3
votes
2answers
105 views

Guessing the other root to a quadratic equation

I just attempted to do the question below, but it seems that even after seeing the answer I'm not sure I understand the motivation for the solution. Let $\alpha ...
0
votes
1answer
55 views

Calculate the sum $\sum^{\infty}_{j=1}(2\sqrt{2}-3)^j$

Would appreciate if anyone could help with the summation \begin{equation*} \sum^{\infty}_{j=1}(2\sqrt{2}-3)^j. \end{equation*} Thanks a lot.
1
vote
0answers
28 views

Polynomial With Complex Zeros

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x) = x^3 - ax^2 + bx - 65$. For each possible combination of $a$ and $b$, let ...
1
vote
1answer
9 views

Sequence of functions that extends the algebraic properties of exponents to higher level operators.

I was thinking about some simple algebraic exponent properties such as the following $$ z^{x+y} = z^xz^y $$ and I started wondering about analytically continuing this identity to "higher-level ...
0
votes
4answers
53 views

Problem Verifying Two Challenging Trig Identities

My math teacher gave us an equality involving trigonometric functions and told us to "verify" them. I tried making the two sides equal something simple such as "1 = 1" but kept getting stuck. I would ...
1
vote
3answers
48 views

Use quadratic formula to find upper and lower limits of an expression

Using quadratic formula show that $\frac{x^2-x+1}{x^2+x+1}$ lies between $3$ and $\frac{1}{3}$ for all real values of $x$. Let $\frac{x^2-x+1}{x^2+x+1}=y$, then ...
1
vote
2answers
38 views

Solving $x^y = y^x$ analytically in terms of the Lambert $W$ function

I'm interested in deriving the solution for $y$ in terms of $x$ given $x^y = y^x$ using the Lambert $W$ function. Wolfram Alpha states: $$y = - \frac{x\ W\left(-\frac{\log(x)}{x}\right)}{\log(x)}$$ ...
0
votes
2answers
28 views

If $\sin s=-1/3$ and $s$ is in the $4$-th quadrant, find the exact value of $\sin (2s)$ [on hold]

Could someone solve this step by step so I can wrap my head around the process?? If $\sin s=-1/3$ and $s$ is in the $4$-th quadrant, find the exact value of $\sin (2s)$.
0
votes
1answer
17 views

Partition set of $n$ elements until each partition contains $1$ element. Must terminate after exactly $n-1$ iterations?

Suppose I have a set of $n$ elements and I want to partition the set (split into two) until each partition contains a single element. How do I see that the terminating case must occur after exactly ...
2
votes
1answer
40 views

Prove that $1/(\sin x + 1) - 1/(\sin x - 1) = 2 \sec^2 (x)$

Can anyone solve this for me? Prove that $\frac1{\sin x + 1} - \frac1{\sin x - 1} = 2 \sec^2 (x)$. This is as far as I went: $$\frac{(sin x - 1) - (sin x + 1)}{(sin x + 1)(sin x - 1)}$$ ...
-3
votes
1answer
21 views
-1
votes
2answers
46 views

Show that if x>0, x+1/x >= 2. [on hold]

How am I to prove this inequality without use of calculus: for any real x>0, x+1/x >= 2 ? Thanks for any help.
0
votes
3answers
55 views

Prove that $\alpha$ lies between $0$ and $4$.

Let $a,b,c$ be the length of the sides of the triangle $ABC$ . Given $(a+b+c)(b+c-a)=\alpha bc$.Then Prove that the value of $\alpha$ lies in between $0$ and $4$. ...
0
votes
5answers
71 views

simplify the following rational expression

Simplify the following $$ \frac{x^2-x-2}{x^2-3x} \times \frac{x^2-x-6}{x^2+5x+4} $$ I don't know how to approach it. I tried doing the quadratics first but now I'm stuck after getting $$ ...
2
votes
3answers
50 views

Do asymptotes disprove 0.9 repeating equal 1?

I am in 9th grade and taking geometry. Several of my friends taking pre-calc say that 0.9999... does not equal 1, but is just an asymptote. I have not taken that subject yet and they don't give any ...
-1
votes
4answers
24 views
-3
votes
2answers
34 views

Find minutes when digit sum is 20? [on hold]

When a digital clock reads 3:47, the sum of digits is 14. How many minutes after 3:47 sum of digits will be 20 for first time? a) 42 b) 132 c)192 d)251 ...
0
votes
2answers
40 views

Trigonometric double angle formulas problem

I want to simplify the answer to an equation I had to compute, namely, simplifying $\sin^2 (2y) + \cos^2 (2y)$. I know that $\sin^2 (y) + \cos^2 (y) = 1$ but is there anything like that I can use at ...
2
votes
1answer
39 views

Solve $z + z^{-1} = x$ with elementary methods

I want to solve the equation$$z + z^{-1} = x\tag{1}$$ with elementary methods. I know the two solutions to be $$x_{1}=\frac{1}{2}x+\frac{1}{2}\sqrt{x^{2}-4}\qquad\text{or}\qquad ...
5
votes
4answers
129 views

How to solve an equation with $x^4$?

Today, I had this question on a Maths test about Algebra. This was the equation I had to solve: $$(1-x)(x-5)^3=x-1$$ I worked away the brackets and subtracted $x-1$ from both sides and was left with ...
1
vote
1answer
36 views

Finding an nth degree polynomial.

First post here. So I'm having a bit of trouble with the eponymous question type. It's a bit embarrassing, as the problem is almost purely conceptual in nature, and I thought I had basically ...
2
votes
6answers
80 views

Solving $e^\frac1x = x$ non-graphically?

This question has come up twice in different tests and the instructions always point out that it should be solved using a graphic calculator. Fair enough, the answer is ≈ 1.76322...(goes on forever?). ...
1
vote
1answer
16 views

Compound Interest Calculation

In __________ years a sum will double at $5\%$ per annum compound interest. Options given are: a. 15 years 3 months b. 14 years 2 months c. 14 years 3 months d. 15 years 2 months The way to ...
6
votes
3answers
830 views

A supposed to be easy calculus problem

Find the values of $m$ if the line $y=mx+2$ is a tangent to the curve $x^2-2y^2=1$. My working: First we differentiate $x^2-2y^2=1$ with respect to $y$ to get the gradient. We get ...