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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3answers
54 views

Prove that A is invertible if $A^2 - 4A -7I = 0$. [duplicate]

The $2 \times 2$ matrix $A$ satisfies $$A^2 - 4A -7I = 0,$$ where $I$ is the identity matrix. Prove that $A$ is invertible. I'm not sure how to do this. Help would be appreciated.
0
votes
1answer
16 views

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there… [duplicate]

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that Aw = v. Show that ${A}$ is invertible. I'm not sure how to do this.
1
vote
1answer
53 views

Proving that matrix in equation is invertible

The $2 \times 2$ matrix ${A}$ satisfies ${A}^2 - 4 {A} - 7 {I} = {0}$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible. I have tried to solve it like a quadratic, but ...
0
votes
2answers
29 views

Solving $f(x) = x^5 +x + 1 = 0$ with halving the interval / bisection method

Question: Use halving the interval / bisection method to approximately solve: $$f(x) = x^5+ x + 1 = 0$$ with a precision of $\pm 0.1$ Attempted solution: The general idea, as I understand it, is ...
0
votes
2answers
14 views

Complex number (Rhombus)

Given that $z_1=1+2i$ and $z_2=\frac{3}{5}+\frac{4}{5}i$, write $z_1z_2$ and $\frac{z_1}{z_2}$ in the form $p+iq$, where $p$ and $q \in R$. In an Argand diagram, the origin O and the points ...
-1
votes
1answer
20 views

Argument of Complex Number (Am I wrong?)

I'm given $z=-2+\sqrt{3}i$. So I worked out the argument of $arg(z)=\tan^{-1}(\frac{\sqrt{3}}{-2})$. I got the answer $2.256$rad. But the given answer is $2.45$rad. Am I wrong?
-1
votes
1answer
26 views

Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
2
votes
3answers
26 views

Absolute Value Inequality Problem

Problem: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals My attempt using the Definition of Modulus: $$\dfrac{2-|x-4|}{|x-4|}>0$$ $$CASE A:x-4\ge 0\Rightarrow x\ge4\Rightarrow ...
1
vote
1answer
33 views

Simplify the expression and find the minimum value

I want to simplify the expression \begin{equation} A = \frac{\sqrt{1-\sqrt{1-x^2}}\Big[\sqrt{(1+x)^3} + \sqrt{(1-x)^3} \Big]}{2-\sqrt{1-\sqrt{1-x^2}}} \end{equation} and find the minimum value of ...
1
vote
1answer
26 views

On finite sums and products

I'd like to get a good book on finite summations and products before I study infinite series more in depth next year. The book should cover geometric/ harmonic sums and prove different formulas for ...
1
vote
0answers
26 views

Calculating a formula for variables with multiple values equaling the same total

I'm having a bit of trouble puzzling a formula for some code I'm using to develop a piece of software. I'm not very savvy with what the technical terms for all of what I'm describing are, but I'll try ...
0
votes
3answers
39 views

Complex number $\tan \alpha+i$

Given that $z=\tan \alpha+i$, where $0<\alpha<\frac{1}{2}\pi$ Find $\left |z \right |$. I've never seen this kind of example in my book. Can anyone guide me? Thanks a lot. How to find $arg ...
0
votes
1answer
25 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 $$ ...
3
votes
2answers
174 views

Square roots of Complex Number. [duplicate]

Calculate, in the form $a+ib$, where $a,b\in \Bbb R$, the square roots of $16-30i$. My attempt with $(a+ib)^2 =16-30i$ makes me get $a^2+b^2=16$ and $2ab=−30$. Is this correct?
0
votes
1answer
23 views

Find the number of possible values of $a$

Positive integers $a, b, c$, and $d$ satisfy $a > b > c > d, a + b + c + d = 2010$, and $a^2 − b^2 + c^2 − d^2 = 2010$. Find the number of possible values of $a.$ Obviously, factoring, ...
2
votes
4answers
60 views

Trigonometric equation $\sec(3\theta/2) = -2$ - brain dead

Find $\theta$ with $\sec(3\theta/2)=-2$ on the interval $[0, 2\pi]$. I started off with $\cos(3 \theta/2)=-1/2$, thus $3\theta/2 = 2\pi/3$, but I don't know what to do afterwards, the answer should be ...
2
votes
2answers
25 views

Ratios and percents

Mrs. Smith has 80 birds: geese, hens and ducks. The ratio of geese to hens is 1:3. 60% of the birds are ducks. How many geese does Mrs. Smith have? a) 16 b) 8 c) 12 d) 11 I know that if 60% if ...
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
46 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
97 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 ...
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 ...
4
votes
1answer
37 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?
2
votes
2answers
52 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 ...
15
votes
9answers
856 views

Computing irrational numbers

I am genuinely curious, how do people compute decimal digits of irrational numbers in general, and $\pi$ or nth roots of integers in particular? How do they reach arbitrary accuracy?
-13
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0answers
39 views

Mathematics question [on hold]

$110$ monkeys run at $3\frac{\text{km}}{\text{sec}}$ for $3$ minutes. How much distance the group has covered? 100m 150m 240m 16500m
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
33 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 ...
0
votes
0answers
43 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 ...
3
votes
0answers
45 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 ...
1
vote
7answers
122 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 ...
-2
votes
1answer
32 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
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?
-2
votes
0answers
58 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
69 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
49 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
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 ...
0
votes
3answers
70 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
113 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
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
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
26 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
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
80 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
21 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
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
26 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 ...