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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9
votes
3answers
437 views

FoxTrot Bill Amend Problems

So I found this on the Wolfram website today: So I was wondering about how one might be able to (if possible) solve those four problems by hand. Here are the problems, $\LaTeX$ed: $ \lim_{x \to ...
1
vote
3answers
32 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 - ...
0
votes
5answers
63 views

How do you factor a quadratic expression, without using the formula?

I am asked to factor $2x^2 -3x+1=0 $ using factorization, but I run into fractions, and it becomes very messy and complicated to deal with, especially since specifically asked not to use the formula. ...
2
votes
2answers
55 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 ...
4
votes
1answer
110 views

Find zeros of function $f(x)$

If I have $$x^2(x-3)(x+3)=0$$ then the solutions are: $$x_{1,2}=0, x_3=3, x_4=-3 $$ or $$x_{1}=0, x_2=3, x_3=-3?$$ So are there 4 or 3 solutions?
3
votes
2answers
88 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
2answers
48 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 ...
0
votes
1answer
28 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{ ...
6
votes
3answers
156 views

Series Question: $\sum_{n=1}^{\infty}\frac{1}{16n^2-1}$

How to compute the following series: $$\sum_{n=1}^{\infty}\frac{1}{16n^2-1}$$ I tried to use partial fraction ...
3
votes
5answers
105 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
0answers
26 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
29 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 ...
0
votes
0answers
35 views

Are there smarter ways to evaluate expressions involving roots? [on hold]

Let's say you want to evaluate something like $$ \frac{\sqrt{5}}{2-\sqrt{5}}+(1+4\sqrt{5})(1-2\sqrt{5}) $$ One can rationalize the denominator and simplify the expression. In this case the answer ...
3
votes
2answers
88 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 ...
1
vote
4answers
72 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 ...
1
vote
1answer
54 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
votes
1answer
29 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$(z)$ ...
2
votes
2answers
55 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
0answers
19 views

General Form Radical Inequality

$±\sqrt{P^n (x)} ± \sqrt{Q^n (x)} < ± \sqrt{R^n (x)} ± \sqrt{S^n (x)}$ I am having a tedious time proving this inequality for the four polynomials of the nth degree. I have squared each side at ...
2
votes
0answers
129 views

Closed formula for the numbers of the form $\sqrt{1+\sqrt{4+\sqrt{9}}}$

how can i find the formula for the nth term of this series? SQ = square root $\sqrt{1} = 1$ $\sqrt{1 +\sqrt{4}} = \sqrt{3}$ $\sqrt{1 +\sqrt{4+\sqrt{9}}} \approx 1.909385061$ $\sqrt{1 ...
5
votes
5answers
183 views

The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
7
votes
2answers
82 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 = $0$?
3
votes
5answers
110 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 + ...
0
votes
5answers
36 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 ...
1
vote
2answers
42 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
3answers
63 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 ...
0
votes
3answers
34 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, ...
1
vote
3answers
47 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.
8
votes
3answers
114 views

Direct formula for area of a triangle formed by three lines, given their equations in the cartesian plane.

I read this formula in some book but it didn't provide a proof so I thought someone on this website could figure it out. What it says is: If we consider 3 non-concurrent, non parallel lines ...
2
votes
1answer
58 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 ...
0
votes
1answer
43 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
41 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, ...
2
votes
1answer
43 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
4
votes
1answer
47 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
3answers
55 views

Simplifying radical expressions such as $\sqrt{80}$

I am having trouble simplifying a radical expression, such as say...$\sqrt{80}$. What I do is firstly, I do 80/2, then 80/3, then 80/4, then 80/5...etc until I find the largest number that can be ...
1
vote
3answers
42 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
vote
4answers
31 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 ...
5
votes
2answers
383 views

How prove this $x+y=0$ if $\left(\sqrt{y^2-x^3}-x\right)\left(\sqrt{x^2+y^3}-y\right)=y^3$

Question: let $x,y$ are real numbers,and such $$\left(\sqrt{y^2-x^3}-x\right)\left(\sqrt{x^2+y^3}-y\right)=y^3$$ show that $$x+y=0\tag{1}$$ before I have solve following problem: if ...
0
votes
1answer
14 views

Factory producing parts efficiency increase and work problem

A certain number of small parts need to be produced. 30 parts are scheduled to be produced after each day. After 1/3 of the parts are produced, the rate of production increases by 10% thanks to ...
2
votes
2answers
27 views

If $ \sum_{r=1}^{13}\frac{1}{r} = \frac{x}{13!}\;,$ Then the Remainder when $x$ is Divided by $11$

If $\displaystyle \sum_{r=1}^{13}\frac{1}{r} = \frac{x}{13!}\;,$ Then the Remainder when $x$ is Divided by $11$. $\bf{My\; Try::}$ Given $\displaystyle \sum_{r=1}^{13}\frac{1}{r} = ...
3
votes
3answers
81 views

Evaluating the sum $1\cdot 10^1 + 2\cdot 10^2 + 3\cdot 10^3 + \dots + n\cdot 10^n$

How can I calculate $$1\cdot 10^1 + 2\cdot 10^2 + 3\cdot 10^3 + 4\cdot 10^4+\dots + n\cdot 10^n$$ as a expression, with a proof so I could actually understand it if possible?
-1
votes
2answers
55 views

Factoring the sum or difference of two cubes

I'm learning about sums and differences of cubes and I can't understand it very well. I am faced with this problem: $$x^3 - 27$$ I am told to find the sum or difference of the two cubes. I ...
-1
votes
1answer
19 views

Finding the value of a variable present in two functions

I have two functions each containing a variable besides an $x$. $$kx-3\quad \text{ and }\quad x^2+k$$ I set them equal to each other, but my algebra is failing me and I can't remember how to solve ...
8
votes
9answers
2k views

What's wrong with solving absolute value equations in this way?

Say I have $3x-2 = |x|$. Why can't I just do this: $3x - 2 = -x$ and $3x - 2 = x$ and then get two values for $x$: $1$ and $0.5$? I know the answer $0.5$ doesn't work if you plug this in. However, I ...
2
votes
3answers
54 views

Ambigous question regarding how to view surds with numbers infront

Say I want to multiply 2 by 5$\sqrt3$ . Do I firstly do 2 * 5, then 2 * 3? I'm not sure about the order of operations here. Such a dumb question, I know. Edit - can someone show me the systematic ...
3
votes
3answers
91 views

How to prove that $\frac{a+b}{2} \geq \sqrt{ab}$ for $a,b>0$?

I am reading a chapter about mathematical proofs. As an example there is: Prove that: $$(1) \space\space\space\space\space\space\space\space\space\space\space \frac{a+b}{2} \geq \sqrt{ab}$$ for ...
2
votes
5answers
2k views

When will these two trains meet each other

I cant seem to solve this problem. A train leaves point A at 5 am and reaches point B at 9 am. Another train leaves point B at 7 am and reaches point A at 10:30 am.When will the two trains ...
-2
votes
2answers
74 views

How to solve the following equation by extracting square roots? [closed]

How to solve this equation by extracting square roots? $$9x^2=36$$ $$(x-12)^2=16$$ $$(x-5)^2=25$$ $$(x+2)^2=14$$ $$(4x+7)^2=44$$ $$(x+5)^2=(x+4)^2$$
1
vote
1answer
52 views

Order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$

There is a multiple choices which says what is the order of $\{x\in\mathbb {Z}, |x|+|3x-1|<5\}$? a. 1 b. 3 c. 2 d. empty I know that by considering certain cases, for example when $x<0$ or ...
-1
votes
2answers
49 views

Squared binomial paradox?

When you square this $$(5-2)^2$$ you will get 49 $$ 5^2 - 2 * 5 * (-2) + (-2)^2$$ $$25 + 20 + 4 = 49$$ but if you do it like this (5-2) * (5-2) you will get 9 $$ 5(5-2) - 2(5-2)$$ $$25-10-10+4$$ ...