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
votes
1answer
26 views

Quick Factoring/Multiplying with recursion question

I am wondering if anyone can help to shed some light on something that I think should be very easy but I dont quite understand. In my textbook, how does author make this conclusion, From $$ ...
0
votes
4answers
112 views

How can I factor the polynomial $125x^3 + 216$?

$$125x^3 + 216$$ I have tried to factor it but because the square root of $216$ is a decimal, I can't figure out how to do the problem.
0
votes
1answer
34 views

Question regarding a loan from a bank for my friend

I have a question please, I'm having a difficult times calculating the right way this one... I took a loan on July 3 2014 of \$50,000 from my bank for my friend, split it on 12 installments (about ...
1
vote
7answers
123 views

How to solve $12-\sin(\theta)=\cos(2\theta)$?

$$12-\sin(\theta)=\cos(2\theta)$$ What's the correct answer on the $[0,2\pi]$? I started with $12-\sin(\theta)=1-2\sin^2(\theta)$ and then i cant get anything sensible as i end up with ...
0
votes
5answers
44 views

How does this seemingly-trivial simplification work?

In a section on inductive proofs in the book Modelling Computing Systems: Mathematics for Computer Science (Muller, Struth) there is a simplification that is assumed to be trivial, but that I can't ...
3
votes
1answer
48 views

weird trig problem $\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \leq 2\pi$

$\tan(\theta)=-\sqrt{2}\sin(\theta)$ on the interval $0 \leq \theta \lt 2\pi$ I started off with $[(\sin(\theta)/\cos(\theta)] \times (1/\sin(\theta) )= - \sqrt 2$, then after simplification i got ...
2
votes
2answers
99 views

Understanding why $a+b\sqrt {2}\neq \sqrt {3} $

I want to intuitively understand why $a+b\sqrt {2}\neq \sqrt {3} $ for $a, b \in \mathbb Q $ I really have no intuition regarding this matter, and have to deal with similar concepts regularly while ...
4
votes
1answer
41 views

Find the range of a $4$th-degree function

For the function $y=(x-1)(x-2)(x-3)(x-4)$, I see graphically that the range is $\ge-1$. But I cannot find a way to determine the range algebraically?
1
vote
1answer
93 views

After Exponentials

Look at this- $$f(a,b)=a+b$$ The next step would be to make a function $g$ such that $$g(a,b)=\underbrace{a + a + a \cdots}_{b\text{ times}}=a\cdot b$$ Then we made $h$ so that ...
2
votes
2answers
54 views

$x^n + y^n = z^n$, $n>1$ To show that $x,y,z$ is greater than $n$

Problem: If $x$,$y$,$z$ and $n>1$ are natural numbers with $$x^n+y^n = z^n$$ then show that x,y and z are all greater then $n$. My approach, from Fermat's Theorem we know that $x^n + y^n = z^n$ ...
0
votes
0answers
38 views

Sum of zeros polynomial

There are nonzero integers $a$, $b$, $r$, and $s$ such that the complex number $r+si$ is a zero of the polynomial $P(x)={x}^{3}-a{x}^{2}+bx-65$. For each possible combination of $a$ and $b$, let ...
0
votes
2answers
115 views

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

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
20 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
35 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 ...
-2
votes
1answer
32 views

How many of each kind? [closed]

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
44 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?
-3
votes
0answers
60 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
71 views

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

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
2answers
114 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 ...
-3
votes
1answer
53 views

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

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
1answer
298 views

Reverse a formula with codependent expressions

I apologize for my title, but I really am a long way from understanding how to even describe my problem accurately, let alone solving it. I'm looking to reverse this formula: ...
0
votes
3answers
73 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
63 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 ...
2
votes
4answers
1k views

Factoring Cubic Equations

I’ve been trying to figure out how to factor cubic equations by studying a few worksheets online such as the one here and was wondering is there any generalized way of factoring these types of ...
1
vote
1answer
38 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
24 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?
2
votes
3answers
40 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
81 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!
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
votes
2answers
21 views

Pricing call options with binomial trees (proof) [closed]

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 - ...
1
vote
0answers
39 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 ...
1
vote
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
32 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
28 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 ...
0
votes
5answers
76 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 $$ ...
0
votes
0answers
22 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 ...
-6
votes
3answers
64 views

Solve the equation $2xy+2x-5y=40$, if $x$and $y$ are whole numbers. [closed]

Solve the equation $2xy+2x-5y=40$, if $x$ and $y$ are whole numbers. Could anyone give me a step by step answer?
0
votes
1answer
57 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
3answers
43 views

Volume of a parallelepiped when not given values for three vectors

There is a parallelepiped determined by three dimensional vectors x, y, and z. The volume of this parallelepiped is $11$. What is the volume of the parallelepiped determined by the three dimension ...
1
vote
1answer
11 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
57 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
0answers
39 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
3answers
53 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 ...
0
votes
3answers
48 views

The inequality $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$

How one can prove the following statement: $k(n-1)<n^2-2n$ for all odd $n$ and $k<n$ Tried so far: induction on $n$, graphing, and rewriting $n^2−2n$ as $(n−1)^2−1$.
1
vote
2answers
41 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: $$y = - \frac{x\ W\left(-\frac{\log(x)}{x}\right)}{\log(x)}$$ ...
6
votes
3answers
840 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 ...
0
votes
2answers
28 views

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

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 $\sin (2s)$.
-3
votes
2answers
34 views

Find minutes when digit sum is 20? [closed]

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 ...
2
votes
1answer
41 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: $$\frac{(sin x - 1) - (sin x + 1)}{(sin x + 1)(sin x - 1)}$$ ...