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
2answers
36 views

Expand the Expression: $\log\left({(x^2+4)^{1/2}}/\left\{(x^2+1)(x^3 - 7)^2\right\}^{1/2}\right)$

Expand the expression: $$\log\frac{\sqrt{x^2+4}}{\sqrt{(x^2+1)(x^3-7)^2}}$$ I got $$\frac{1}{2}\biggl(\log(x^2+4)-\log(x^2+1)+2\log(x^3-7)\biggr)$$ but the answer key says: ...
3
votes
5answers
165 views

Factoring $(x+1)(x+2)(x+3)(x+4)+1$

I have to factor this polynomial: $$(x+1)(x+2)(x+3)(x+4)+1$$ WolframAlpha gives $$(x^2+5x+5)^2$$ How can I prove it without expanding the result? Thanks!
0
votes
3answers
186 views

Do you flip the inequality sign if multiplying a quadratic equation by $-1$?

$$(-1)(-x^{ 2 }+3x+18)<0(-1)$$ becomes $$x^{ 2 }-3x-18>0\quad ?$$ I want to confirm before proceeding in solving a quadratic inequality.
0
votes
1answer
36 views

Unable to figure out why the solution to a quadratic inequality works

$$x(2x+7)\ge 0$$ I solved this inequality a few days ago and I was referencing it to complete other ones just now and am at a loss for how I came up the with solution. $$(-\infty ,-\frac { 7 }{ 2 } ...
0
votes
2answers
30 views

Why is my answer always coming out positive?

My answer is always 3, but the correct answer is negative 3. What I do is multiply everything by 10 because the common factor between all THREE components is 10. Then I cancel the 10 with the ...
0
votes
3answers
58 views

Why is $2\sqrt5$ equal to $\sqrt{20}$?

What does the 2 in front mean? Is it like root 5 * root 5? It can't be because the answer doesn't numerically evaluate to 2root5. Just looking for some intuition here because I am reading the numbers ...
3
votes
3answers
97 views

How to solve? $3A^2 - 2A^3 = 1.60$ [closed]

How do I solve this? I'm not good enough to solve this. Sorry! $$ 3A^2 - 2A^3 = 1.60 $$
0
votes
3answers
101 views

Why does this work for $ i^{2i} $?

I'm finding the principal value of $$ i^{2i} $$ And I know it's solved like this: $$ (e^{ i\pi /2})^{2i} $$ $$ e^{ i^{2} \pi} $$ $$ e^{- \pi} $$ I understand the process but I don't understand ...
8
votes
8answers
5k views

How should I self-study calculus? [duplicate]

So I already took Pre-Calc, and ended up with a B both semesters. I am an incoming senior in high school. My special-ed case manager won't let me take it because she doesn't want to see me panic ...
1
vote
7answers
363 views

How to solve the quadratic equation $x^2-1=2$?

Solve $x^2-1=2$ I have no idea how to do this can somebody please help me? I have tried working it out and I could never get the answer.
0
votes
1answer
41 views

How to divide $6^{4/6}$ by $6^{6/8}$?

How do you resolve this: base 6 and exponent 4/6 divided by base 6 and exponent 6/8?
4
votes
2answers
227 views

Expressing the area as a function :)

Express the area A of an equilateral triangle as a function of the height of the triangle. Thanks :) I am not sure where to even start on how to answer this problem.
0
votes
4answers
94 views

Is $\frac{4x + 2}{12 x ^2}$ simplifiable?

I'd like to know what methods can I apply to simplify the fraction $\frac{4x + 2}{12 x ^2}$ Is it valid to divide above and below by 2? (I didn't know it but Geogebra's Simplify aparantly does ...
0
votes
1answer
126 views

rationalize numerator and find conjugate

So in our math class today we were going over some problems on how to rationalize the numerator and finding the conjugates. I am completely lost on how you are supposed to go about it I could really ...
2
votes
3answers
86 views

How do I simplify $\frac{\sqrt{21}-5}{2} + \frac{2}{\sqrt{21} - 5}$?

How do I simplify the following equation? $$\frac{\sqrt{21}-5}{2} + \frac{2}{\sqrt{21} - 5}$$ I have no idea where to start. If I multiply either fraction by its denominator I will still end ...
2
votes
1answer
50 views

Find area of a magnified polygon

I have a highly irregular polygon made out of polylines for which I KNOW the area A1, the perimeter P1. I can offset this polygon (magnify it) by offsetting every site by a given distance d. Is there ...
0
votes
2answers
33 views

Problems with rationalizing an expression

I am faced with this problem: Express $R$ as a rational expression: $\frac1{R} = \frac1{R_1} + \frac{1}{R_2} + \frac1{R_3}$ If I'm correct, rationalizing an expression means simplifying it. I ...
2
votes
1answer
50 views

An inequality for a quotient of polynomials

I am trying to prove the following to be true for $n > 1$: $$\frac{n^4}{n^3 + 1} \le Cn$$ It seems like there is some basic rule where you multiply the 1 in the denominator by a value which makes ...
2
votes
6answers
87 views

Why is the result of $-2^2 = -4$ but $(-2)^2 =4$?

I am really new into math, why is $-2^2 = -4 $ and $(-2)^2 = 4 $?
2
votes
3answers
294 views

The mth term of a Geometrical Progression is n and nth term is m. Find (m+n)th term

The mth term of a Geometrical Progression is n and nth term is m. Find (m+n)th term. I've tried this: Tm = arm-1 = n (Eq 1) Tn = arn-1 = m (Eq 2) Subracting 2 from 1 rm - r - rn + r = n-m rm - ...
1
vote
1answer
383 views

Find distance between two stations given travel times of trains

Problem Two trains A and B start from two points P1 and P2 respectively at the same time and travel towards each other. The difference between their speed is 10 kmph and train A takes one hour more ...
2
votes
2answers
1k views

Alternating sum of binomial coefficients is equal to zero [duplicate]

Prove without using induction that the following formula:$$\sum_{k=0}^n (-1)^k\binom{n}{k}=0$$ is valid for every $n\ge1$. Progress For each odd $n$ we can use the ...
1
vote
0answers
52 views

I want to know how the function $f(x) = x^2$ can be scaled into$ f(x) = 3x - 3x^2 - 1/3$

I want to know the horizontal and vertical scaling that transforms the equation $$ y = x^2 $$ into $$ y = 3x - 3x^2 - 1/3 $$ and please show calculations.
2
votes
2answers
102 views

How to express $\log_2 (\sqrt{9} - \sqrt{5})$ in terms of $k=\log_2 (\sqrt{9} + \sqrt{5})$?

If $$k=\log_2 (\sqrt{9} + \sqrt{5})$$ express $\log_2 (\sqrt{9} - \sqrt{5})$ in terms of $k$.
0
votes
1answer
32 views

Trying to calculate 5 simultanious equations in Mathematica

$\def\1{x_1}\def\2{x_2}\def\3{x_3}\def\f{f(\1,\2,\3)}\def\bs{\bigskip}\def\b{\begin{pmatrix}}\def\e{ ...
0
votes
2answers
57 views

Pay back loan with an annual withdraw

I was given question 7b as homework: I am guessing that there are numerous ways of approaching this. The one method I have tried was to calculate the effective interest year for the year. Then ...
1
vote
0answers
42 views

Find all ordered pairs $(a, b)$ in $a+\frac{10b}{a^2+b^2} = 5\;\;,b+\frac{10a}{a^2+b^2}=4$

Find all ordered pairs $(a, b)$ of complex numbers with $a^2+b^2\neq 0,$ and $\displaystyle a+\frac{10b}{a^2+b^2} = 5\;\;,b+\frac{10a}{a^2+b^2}=4$ $\bf{My\; Solution}::$ Using Complex ...
0
votes
2answers
151 views

Modeling with equations riddle

A father said that sevens years ago, he was eleven times as old as his daughter. Now he is four times as old as she is. How old is the father? Can this be solved as a system of equations? I am stuck ...
2
votes
5answers
119 views

How does $2^n + 2^n = 2^{n+1}$?

What property of exponents can be used to show that $$2^n + 2^n = 2^{n+1}$$ Does this work for all constants raised to a variable exponent?
1
vote
1answer
102 views

Polar form of the sum of complex numbers $\operatorname{cis} 75 + \operatorname{cis} 83 + \ldots+ \operatorname{cis} 147$

The number $\operatorname{cis} 75 + \operatorname{cis} 83 + \operatorname{cis} 91 +\dots+ \operatorname{cis} 147$ is expressed in the form $r\operatorname{cis}(\theta)$, where $0\leq \theta< ...
1
vote
4answers
89 views

Solve $\log_4 ( 16^{100})$

How do i evaluate $$\log_4 { 16^{100}}$$ After finding the $\log_4$ of $16$ which is $2$, how do I get $200$ and why? Wouldn't the $2$ be squared by $100$? Or wouldn't the $2$ be on the other side ...
3
votes
2answers
479 views

Find the number of escalator steps from the number of steps made by people walking on it

Renata walks down an escalator that moves up and counts $150$ steps. Her sister Fernanda climbs the same escalator and counts $75$ steps. If the speed of Renata (in steps per time unit) is three times ...
2
votes
2answers
112 views

What is the meaning of this Wolfram Alpha result when calculating $3^p = 4^q$?

I would like to know are the some $p \in \mathbb{N}$ and $q \in\mathbb{N}$ for $3^p = 4^q$ except the trivial $p = q = 0$. So, I entered the expression into Wolfram Alpha, which returned the result ...
-1
votes
1answer
41 views

Polar form of complex numbers3

Let $z$ be the complex number $z=-2+i$ and let the angle $\phi$ be such that tan$\phi=1/2$ and $-\pi/2<\phi<-\pi/2$. Calculate the modulus $|z|$ and describe the principal argument arg$(z)$ ...
7
votes
2answers
482 views

Is $f(x)=10$ a periodic function?

I am not getting satisficatory explanation for this. Clearly $f(x+T) = f(x)$ for all values of $T$. If we assume it is periodic, does this mean period = $0$?
2
votes
2answers
84 views

Using factoring to solve the equation $(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$

Solve for all values of $r$: $$(r^2 + 5r - 24)(r^2 - 3r + 2) = (4r - 10)(r^2 + 5r - 24)$$ I'm not sure how my thinking isn't really correct here. I know this all seems very elementary and such, ...
0
votes
5answers
4k views

Raising a number to a negative fraction power

I am doing a math problem where I need to raise 9 to the -3/2 power. I am unsure how this is done. I believe it's the equivalent of saying 2√9^-3, but I am unsure if this is true. If you can help me ...
0
votes
3answers
108 views

Modeling with an equation

The fish population in a lake rises and falls according to the formula $$F=1000(30+17t-t^2)$$ Here $F$ is the number of fish at the time $t$, where $t$ is measured in years since January 1, 2002, ...
0
votes
2answers
59 views

Factoring Questions

I have to complete a factoring packet for AP Calculus, and I'm having trouble with three of the questions... Find the missing factor: 1. $2\sqrt{x} + 6x^\frac 32 = 2\sqrt{x}$(_____________) ...
2
votes
5answers
126 views

Dividing by $\sqrt n$

Why is the following equality true? I know I should divide by $\sqrt n$ but how is it done exactly to get the RHS? $$ \frac{\sqrt n}{\sqrt{n + \sqrt{n + \sqrt n}}} = \frac{1}{\sqrt{1 + ...
-2
votes
3answers
77 views

Algebra-Precalculus Questions

1. (A graphing question) $f(x) = \begin{cases} 0, & \text{if $x$ is rational} \\ 1, & \text{if $x$ is irrational} \end{cases}$ I'm not exactly sure how to graph this. I'm thinking that it ...
1
vote
3answers
72 views

An equation involving fractional powers

How would I solve: $${ x }^{ \frac { 2 }{ 3 } }=2$$ I am at the last part of solving an equation of the quadratic type and got stuck here.
2
votes
1answer
70 views

$e^x$ defined $a^x$

I have read the chapter up and down but I do not see how, I would like to not take anything from the book but start on e fresh example as I think that would help me to realise what is going on. Im ...
1
vote
3answers
59 views

Quadratic formula and factoring are leading to different answers

$$x^{ 2 }-2x-15=0$$ By factoring, I get: $$(x-5)(x+3)$$ Which has the solutions: $$x=5, x=-3$$ However when I use the quadratic formula (which is what the book saids to use), I get $$\frac { 2 ...
-1
votes
2answers
68 views

Showing that the roots of an equation are real and distinct.

I need help showing that the roots of the equation $px^2 - 3x - p = 0$ are real and distinct for all real values of $p$. After some help from the previous answers, I am now stuck on $4p^2 + 9 > ...
0
votes
1answer
46 views

Understanding of $\frac{d\text{Ln}(x)}{dx}$

I am looking into my textbook, the calculations done here are simple to follow so that is not my question, my question is is more the understanding of why they have used h in the numerator of the ...
0
votes
1answer
48 views

How to solve this linear system using determinants and using matrices

$$\left\{\begin{array}{c} 2x + y + 3z = 1 \\ 5x + z = 3y - 3 \\ 2y + z = 4 \\ \end{array}\right.$$ Here is my problem. I know how to calculate it using matrices, but i don't know how to organize it, ...
4
votes
1answer
62 views

Where did I go wrong in completing the square?

$$2x^{ 2 }+8x+1=0$$ Move 1 to the other side of the equation: $$2x^{ 2 }+8x\quad =-1$$ Divide both sides by 2 to get 1 as the leading coefficient: $$x^{ 2 }+4x\quad =-\frac { 1 }{ 2 } $$ ...
1
vote
4answers
39 views

Were exactly did I go wrong in rationalizing denominator?

The question is to rationalize: $\frac{\sqrt5}{\sqrt10 - \sqrt5}$ I stopped at $\sqrt50 + 1$ after multiplying by the conjugate and cancelling out everything because I knew at this point my answer ...
2
votes
1answer
65 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$