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.

learn more… | top users | synonyms (2)

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 ...
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 ...
3
votes
5answers
106 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
56 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
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 ...
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 ...
3
votes
2answers
81 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
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$(z)$ ...
7
votes
2answers
83 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
56 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
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 ...
0
votes
3answers
35 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
2answers
43 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. $2\sqrt{x} + 6x^\frac 32 = 2\sqrt{x}$(_____________) ...
3
votes
5answers
113 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
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 ...
1
vote
3answers
48 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
59 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
43 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
54 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
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, ...
4
votes
1answer
48 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
32 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
43 views

Sum involving integer part and cosine function

How to find the close form of sum and eliminate $k$? $$ \sum_{k=1}^{n} \frac{n \left[ \cos \left( \frac{n}{k}- \left[\frac{n}{k} \right]\right) \right]}{k} $$
1
vote
3answers
59 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 ...
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} = ...
5
votes
5answers
184 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 ...
3
votes
3answers
83 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
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 ...
-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 ...
2
votes
3answers
56 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 ...
0
votes
0answers
35 views

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

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 ...
-1
votes
2answers
51 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 $$ 5(5-2) - 2(5-2)$$ $$25-10-10+4$$ ...
-2
votes
2answers
77 views

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

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

How to solve the system of equations $y= 5x^2-2x$ and $y=10x+9 $?

I am trying to solve this system of equations $y= 5x^2-2x$ and $y=10x+9$. I have worked the problem out and I am lost could someone please explain it? I have a test tomorrow over this information.
0
votes
2answers
49 views

Factoring $x^3-8$ by grouping

I'm trying to factor by grouping. It worked for me with polynomial division, but I can't get it to work by grouping. $$x^3-8$$ The answer should be $(x−2)(x^2+2x+4)$. So first, the groups are: ...
0
votes
0answers
10 views

what is the best method to solve a large set of sparse nonlinear equation

I am trying to solve a set of nonlinear equations which is big. Indeed, newton method is not recommended since the equatios are sparse and the jacobian is not easy to inverse. Any recommendation for ...
-1
votes
1answer
80 views

Integrating $\int\frac{x^2+1}{(x-1)^3(x+2)}\mathrm dx$ [closed]

I am struggling with the following integral: $$\int\frac{x^2+1}{(x-1)^3(x+2)}\mathrm dx$$ I guess it all comes down to a fairly simple algebraic manipulation - but I cannot see it...
0
votes
1answer
26 views

$\log \left(a^x\right)=x \log (a)$ and $-\log _x(2)=-\frac{1}{\log _2(x)}$

Since $a^x$ has domain $(-\infty ,\infty )$, $\log _a(x)$ has range $(-\infty ,\infty )$. Since $a^x$ has range $(0 ,\infty )$. Since $a^x$ and $\log _a(x)$ are inverse functions, the following ...
0
votes
1answer
29 views

Customers decrease with price increase: find the maximum price before no customers can afford candies.

If the price of the candies is $\$10$ per box, there will be $100$ boxes sold. For each $\$1$ increase in price, $5\%$ of the customers no longer can afford candies. What is the maximum price that can ...
0
votes
2answers
52 views

Is there a nice way to simplify this expression?

Is there a way to get a clean expression for "y" out of this? E.g. "y = ..." $y^q=a y^x+b y^z$ It seems like the most obvious thing would be to take logs, but I was wondering if there are any ...
2
votes
4answers
88 views

Cannot find any excluded values in this fraction

I am learning about excluded values. I am faced with this problem: $$\frac{x}{ x^2 + 9}$$ I started by trying to solve $x$ for $0$ in the denominator: $$x^2 + 9 = 0$$ $$x^2 + 3^2 = 0$$ I then ...
2
votes
4answers
64 views

Show that $ax^2+2hxy+by^2$ is positive definite when $h^2<ab$

The question asks to "show that the condition for $P(x,y)=ax^2+2hxy+by^2$ ($a$,$b$ and $h$ not all zero) to be positive definite is that $h^2<ab$, and that $P(x,y)$ has the same sign as $a$." Now ...
6
votes
4answers
616 views

Solving these two equations simultaneously

I'm having a hard time to solve these two equations simultaneously. I'm arriving to a very long equation.. $$x_0^2+y_0^2=(7\sqrt{2})^2=98$$ $$\sqrt{25+(x_0+2)^2}+\sqrt{4+(y_0-5)^2}=7\sqrt{2}$$
0
votes
1answer
90 views

I can't solve this equation, is it my or book's error?

I found this in my book but whatever I do I can't solve it $$(2x + 3)^2 - (1 + 2x)(2x - 1) = x^2 - (x - 1)^2 $$ it says that the solution is $\{-1.1\}$. Am I doing something wrong or not?
-1
votes
2answers
71 views

How to write these quadratic equation in general form? [closed]

Write the quadratic equation in general form: 1. $x^2=16x$ 2. $13-3(x+7)^2=0$ 3. $x(x+2)=5x^2+1$
0
votes
4answers
28 views

Converting Numbers to Exponential Form

can someone explain to me how problems such as: are converted into exponential form? I am interested in the logic and method behind this, rather than the answer. Much thanks in advance.
2
votes
0answers
133 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 ...