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
3
votes
6answers
75 views
Solving $\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$
Where do I start to solve a equation for x like the one below?
$$\sqrt{7x-4}-\sqrt{7x-5}=\sqrt{4x-1}-\sqrt{4x-2}$$
After squaring it, it's too complicated; but there's nothing to factor or to ...
22
votes
9answers
2k views
Is this way of teaching how to solve equations dangerous somehow?
Two years ago, I bought the book Mathematics for the Nonmathematican, by Morris Kline.
There I learned a new way of solving equations, which is related to the principle that states that any ...
1
vote
1answer
84 views
For which numbers $c$ is there a number $x$ such that $f(cx)=f(x)$?
This is one exercise in Spivak's book that is bugging me for a while, first I thought that $c=1$, but there's a hint:
There are a lot more than you might think at first glance.
And here I'm ...
8
votes
5answers
334 views
solving $\sqrt{3-\sqrt{3+x}}=x$.
Can we solve the following equation in $\mathbb{R}$ without expanding it into a fourth degree equation :
$$ \sqrt{3-\sqrt{3+x}} = x.$$
squaring both sides and squaring again is the only thing I ...
3
votes
3answers
80 views
is $(x+1)^4-x^4$ non-prime for all natural positive integers $x$
Looking at difference between two neighbouring positive integers raised to the power 4, I found that all differences for integer neighbours up to $(999,1000)$ are non-prime.
Does this goes for all ...
0
votes
2answers
28 views
System of inequalities word problem
Andrea's Sequoia gets 12 miles per gallon while driving in the city, and 20 miles per gallon while on the highway. The last time she filled up the car it took 16 gallons of gas, and she had driven 280 ...
3
votes
2answers
100 views
how to solve this multivariate quadratic equation?
Any hope to acquire an analytic solution to such equations:
Solve:
$$\sum_{j=1}^n a_{ij} x_i x_j = b_i$$
for $i=1,\ldots,n$, where $a_{ij}$'s and $b_i$'s are known constants and $x_i$'s are unknowns ...
3
votes
1answer
37 views
Is it possible to rationalize a denominator containing two cube roots?
The fraction in question is
$$-\frac{12}{\sqrt[3]{12\sqrt{849} + 108} - \sqrt[3]{12\sqrt{849} - 108}}$$
And was reached in calculating the solution to $x^4 - x - 1 = 0$. I've tried all the standard ...
1
vote
1answer
22 views
Like term reduction
In finding the derivative of $f(x) = 4x - x^2$ we first find the difference of the numerator $f(x + h) - f(x)$. Therefore we have $f(x + h) = 4(x + h) - (x + h)^2 = 4x + 4h - x^2 - 2xh - h^2$ minus ...
19
votes
11answers
808 views
Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$
Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)?
$$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$
Using the calculator I can see that the first one ...
2
votes
2answers
53 views
How do I solve $x^2 + y^2 + xy = z$ for $y$
How do I solve the following equation for $x$ or $y$ (does not matter because you can swap them):
$$
x^2+y^2+xy=z
$$
3
votes
2answers
155 views
Does this polynomial factorize further?
I just did a national exam and this question was in it; I am convinced this does not work:
Given that $(x - 1)$ is a factor of $x^3 + 3x^2 + x - 5$, factorize this cubic fully.
My attempt
1 | ...
1
vote
1answer
37 views
How do you solve for Y?
I know there has got to be a way to solve for $Y$ but I just can't seem to figure it out. Does anyone know how to solve this? Please help :)
$$5(Y(8))=C$$
$$C(Y(4))=B$$
$$B(Y(2))=A$$
...
3
votes
4answers
67 views
Solve equation $\sqrt{s+13} - \sqrt{7-s} = 2$
Solve the equation
$$\sqrt{s+13}-\sqrt{7-s} = 2$$
I moved the $-\sqrt{7-s}$ to the right side
Thus, I had
$$\sqrt{s+ 13} = 2 +\sqrt{7-s}$$
I then squared both sides
$$\sqrt{s+ 13}^2 = \left(2 ...
3
votes
3answers
71 views
Determining $\sin(15)$, $\sin(32)$, $\cos(49)$, etc.
How do you in general find the trigonometric function values? I know how to find them for 30 45, and 60 using the 60-60-60 and 45-45-90 triangle but don't know for, say $\sin(15)$ or $\tan(75)$ or ...
0
votes
0answers
15 views
Find the terminal point when the distance is not in terms of $\pi$
From Stewart Precalculus 5th edi, P407
I am not sure what to do here, in the textbook, Steward didn't provide any example as to finding the terminal point when the distance $t$ is an integer. I ...
1
vote
1answer
49 views
For which values is $x^3$ less than or equal to $3x$?
The title says it all. The answers say:
$x\le -\sqrt{3}$ and $0\le x\le \sqrt{3}$
(can someone edit this so all the $<$ have an 'or equal to' sign. Edit the roots as well please.
I'm not sure ...
1
vote
1answer
19 views
Number of equivalent rectangular paths between two points
I am trying to determine the number of paths between two points.
I am representing the paths as a list of steps "ruru" = right -> up -> right -> up
For my purposes, we can assume that there will ...
0
votes
0answers
32 views
Quaternion exponential map, rotations and interpolation
A code snippet I need to optimize is performing something peculiar. It seems that it's somehow related to transforming from a frame of reference to another. This is what it does, in mathematical ...
12
votes
3answers
107 views
$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$
Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost.
Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying:
$$x_1 + x_2 \dots + x_k = 0$$
$$x_1^2 + x_2^2 ...
7
votes
5answers
392 views
Solve equations $\sqrt{t +9} - \sqrt{t} = 1$
Solve equation: $\sqrt{t +9} - \sqrt{t} = 1$
I moved - √t to the left side of the equation $\sqrt{t +9} = 1 -\sqrt{t}$
I squared both sides $(\sqrt{t+9})^2 = (1)^2 (\sqrt{t})^2$
Then I got $t + 9 ...
2
votes
0answers
45 views
Is there any easy way to simplify the following term?
$[p\times (p+1)\times (p+2)\times (p+3)\times (p+q)^3] - [4\times p^2\times (p+1)\times (p+2)\times (p+q)^2\times (p+q+3) ] +[6\times p^3\times (p+1)\times (p+q)\times (p+q+2)\times (p+q+3)] -[3\times ...
1
vote
1answer
28 views
Is this enough info to solve this time dilation problem
There are two clocks. One is a regular clock measuring regular time $\tau$. The other is a clock measuring time $t$ which also advances clockwise, but does not advance uniformly--it accelerates ...
1
vote
2answers
64 views
Solving a system of equation:
Solve for $x,y$:
\begin{align}
x^3 + y^3&=2\\
x^2 +x + 9y - 3y^2&=8 \\
\end{align}
I can find $x=y=1$ by guessing. Please help me solve it without using computer.
Thanks
Edited, sorry, I ...
4
votes
2answers
68 views
Trigonometry Airplane question. Finding bearing and distance.
A little background(if you don't care for my story, skip straight to the question): I've missed a few lectures from my teacher because I fell ill. Since I have no information to work with other than ...
-1
votes
1answer
42 views
If 3 out of 5 randomly chosen reports used incorrect style, how many to expect out of 125
A teacher randomly reads 5 written reports from the 125 she has to grade. He finds that 3 reports did not follow the correct style. At this time same rate, how many of the 125 can she expect to not ...
3
votes
1answer
44 views
How to solve these elementary algebra questions?
Lionel is thinking of two numbers. The sum of twice the larger number and 4 times the smaller number is 40. Twice the smaller number is 4 more than larger number. Find the larger number.
Mrs tan ...
0
votes
1answer
252 views
what is the growth rate and continuous growth rate?
The problem: $\;f(t) = 4 \cdot 2^{\,t/5}$
I know continuous is e^k, but this problem doesn't seem to work for that. Is the initial value, 4, able to be used to find the growth rates?
2
votes
4answers
154 views
For $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$ .
If $(x+\sqrt{x^2+3})(y+\sqrt{y^2+3})=3$, compute $x+y$.
2
votes
1answer
25 views
Need help with algebra portion of calculus finding slope of secant line
The example problem is: Given f(x) = $x^2$,find and simplify the slope of the secant line for a = 1 and h = any non-zero number.
The answer is as follows:
For a = 1 and h any non-zero number, the ...
-10
votes
1answer
58 views
Nate's older brother! [closed]
Nate's older brother Mickey has been looking for an old used car just to drive to work. An advertisement in the paper is offering a car listed at $899.95 for sale at 25 percent off. The best estimate ...
-1
votes
3answers
42 views
question about percentages
At the start of the new year, the price for a gallon of milk at a local grocery store increased by 2.4% as a result of inflation. If a gallon of milk cost $2.92 last year, about how much does it cost ...
2
votes
1answer
50 views
what is $\lim_{x\to6}\frac{5}{(x-6)^2}$
$$\lim_{x\to6}\frac{5}{(x-6)^2}$$
Is it undefined or infinity and why ?
1
vote
1answer
60 views
How are these inequalities simplified?
How does this:
(a > b && a > c && b <= c) ||
(a > b && a <= c && b < c)
simplify down to this:
...
1
vote
1answer
39 views
$p(x)$ and $q(x)$ be non-zero polynomials with real coefficients such that…
I am stuck on the following problem:
Let $p(x)$ and $q(x)$ be non-zero polynomials with real coefficients such that
degree$(p(x)) > \text{degree}(q(x))$. If the graphs of $y = p(x)$ and $y = ...
3
votes
2answers
59 views
Solving 3 simultaneous cubic equations
I have three equations of the form:
$$ i_1^3L_1 + i_1K +V_1 + (i_2+i_3+C)Z_n = 0 $$
$$ i_2^3L_2 + i_2K +V_2 + (i_1+i_3+C)Z_n = 0 $$
$$ i_3^3L_3 + i_3K +V_3 + (i_1+i_2+C)Z_n = 0 $$
where $ ...
1
vote
1answer
32 views
What is orthogonal projection of zero to triangle generated by three points (0,1) (5,0) and (2,4)
What is orthogonal projection of zero to triangle generated by three points (0,1) (5,0) and (2,4)
Well, in my opinion, there is none. However, my teacher think that it has but he don't know how ...
1
vote
1answer
40 views
Does There Exist a Term for the Unique Nonpositive Square Root of a Nonnegative Real Number?
The term "principal square root" describes the unique nonnegative square root of a nonnegative real number.
Does there exist a term to describe the unique nonpositive square root of a nonnegative ...
2
votes
3answers
57 views
write an expression for the nth term of the sequence…
write an expression for the nth term of the sequence 0, 7, 16, 27, 40
It is neither geometric or arithmetic
My teacher gave us a key and I can't read his writing...please any help would be great!!
2
votes
4answers
62 views
Solve the equation $\sqrt{3x-2} +2-x=0$
Solve the equation: $$\sqrt{3x-2} +2-x=0$$
I squared both equations $$(\sqrt{3x-2})^2 (+2-x)^2= 0$$
I got $$3x-2 + 4 -4x + x^2$$
I then combined like terms $x^2 -1x +2$
However, that can not ...
3
votes
2answers
55 views
How to define this pattern as $f(n)$
Given a binary table with n bits as follows:
$$\begin{array}{cccc|l}
2^{n-1}...&2^2&2^1&2^0&row\\ \hline \\ &0&0&0&1 \\ &0&0&1&2 \\ ...
0
votes
1answer
33 views
Formula to increase/decrease a relative number based on a fixed number
Being a fairly good math student in high school, this is humbling. But my knowledge about graphs and formulas has greatly diminished since then.
I'm trying to write a formula that calculates a ...
0
votes
1answer
26 views
simplify equation by removing double summation
Is it possible to simplify the following function by removing the double summation?
$$f(x) = \sum_{n=1}^{x-1} \sum_{m=n+1}^x a_{n}b_{m}$$
Or is there no way of removing the sigmas?
Thanks in ...
2
votes
1answer
29 views
Linear independence of $(x\sin x)^{\frac{n-1}{2}}$ and $(x\sin x)^{\frac{n+1}{2}}$
Could you tell me why $(x\sin x)^{\frac{n-1}{2}}$ and $(x\sin x)^{\frac{n+1}{2}}$ are lineraly independent?
I've tried $\alpha(x \sin x)^{\frac{n-1}{2}} + \beta (x\sin x)^{\frac{n+1}{2}} =0$
...
0
votes
0answers
48 views
Matching numbers by $f(x)=\frac{1}{x}$
Let $0<x \leq 1$, We define a function such that $f(x)=y=\frac{1}{x}$ which results $y \geq 1$ . We have infinitely many numbers between $0$ and $1$, so we can match any $x$ to a number $y$ greater ...
0
votes
0answers
25 views
Does $\sum_{k=0}^n x^k = \prod_{k=1}^n \left(x - \mathtt{i}^\frac{2 k}{n+1}\right)$?
This seems to be true:
$$\sum_{k=0}^n x^k = \prod_{k=1}^n \left(x - e^\frac{2\pi i k}{n+1} \right)$$
but I don't know how to demonstrate it, and definitely not neatly. I'd like to see why it should ...
3
votes
3answers
97 views
How to factor a four term polynomial without grouping?
$$2x^3 + 9x^2 +7x -6$$
This equation doesn't factor by grouping, and other than that I have no idea how to solve this problem. Will someone please help?
21
votes
9answers
2k views
Direct Proof that $1 + 3 + 5 + \cdots+ (2n - 1) = n\cdot n$
Prove that for all integers $n$, $n \geq 1$,
$$1 + 3 + 5 + \cdots + (2n - 1) = n\cdot n$$
How would I go about proving this?
2
votes
3answers
61 views
Solve for $x$, $3\sqrt{x+13} = x+9$
Solve equation: $3\sqrt{x+13} = x+9$
I squared both sides and got $9 + x + 13 = x^2 + 18x + 81$
I then combined like terms $x^2 + 17x + 59 = 0$
I then used the quadratic equation $x= -\frac{17}2 ...
1
vote
3answers
92 views
Solving the equation $\dfrac{(1+x)^{36} -1}{x} =20142.9/420$ for $x$.
How would one solve for x in the following equation:
$\dfrac{(1+x)^{36} -1}{x} =20142.9/420$
I tried factorising the top but that didnt really help much.
$((1+x)^{18} - 1)((1+x)^{18}+1)$
Any help ...
