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
2
votes
2answers
1k views
Quartic Equation Solution and Conditions for real roots?
Q1. How to solve a Quartic Equation. There is an online calculator available (and many more similar) that gives the precise answers and also defines the method. Does anyone know what the source of ...
5
votes
1answer
192 views
Find the value of $x^3-x^{-3}$ given that $x^2+x^{-2} = 83$
If $x>1$ and $x^2+\dfrac {1}{x^2}=83$, find the value of the expression$$x^3-\dfrac {1}{x^3}$$
a) $764$
b) $750$
c) $756$
d) $760$
In this question from given I tried to ...
1
vote
1answer
74 views
$\pi$ is just a number, or also the circumference of a sub-unit circle?
A unit circle defined in the Cartesian plane has a radius of $1$ and a diameter of $2$. So making a full round is $2 \pi$. Now, $\pi$ is the ratio of the circumference over the diameter, so if I have ...
0
votes
1answer
22 views
Please help finishing the calculation to find the Entropy of Pareto distribution.
Let $X$ follow Pareto distribution with parameters $\alpha, a, h$. That is, $X\sim Pa(\alpha,a,h)$, where $\alpha>0$ is the shape parameter, $-\infty < a < \infty$ is the location parameter, ...
3
votes
5answers
40 views
Simplify with fractional exponents and negative exponents
I am trying to simplify
$$ \left(\frac{3x ^{3/2}y^3}{x^2 y^{-1/2}}\right)^{-2} $$
It seems pretty simple at first. I know that a negative exponent means you flip a fraction. So I flip it.
$$ ...
26
votes
2answers
891 views
Find all real numbers $x$ for which $\frac{8^x+27^x}{12^x+18^x}=\frac76$
Find all real numbers $x$ for which $$\frac{8^x+27^x}{12^x+18^x}=\frac76$$
I have tried to fiddle with it as follows -
$$2^{3x} \cdot 6 +3^{3x} \cdot 6=12^x \cdot 7+18^x \cdot 7$$
$$ 3 \cdot ...
1
vote
0answers
29 views
Please help finishing the calculation to prove that ” Pareto distribution & Power distribution has inverse relationship”.
Let X follows Pareto distribution with parameters α, a, h.
that is X~Pa(α,a,h)
Where, α>0 is the shape parameter, -∞< a <∞ is the location parameter, h>0 is the scale parameter.
...
2
votes
1answer
21 views
Scale rectangles so they have same height and don't exceed a total width?
I have three rectangles of different sizes side by side.
I want to scale them all (maintaining their aspect ratio) so they have the same height and don't exceed a total width.
I know I could find ...
6
votes
7answers
127 views
Complete the square for $f(x) = 2x^2 + 4x - 6$
I'm studying for a math test. This is the question:
$f(x) = 2x^2 + 4x - 6$. complete the square.
This is how much I get out of the question:
$$2x^2 + 4x - 6$$
$$2(x^2 + 2x - 3)$$
$$2(x^2 + 2x + ...
2
votes
3answers
58 views
Series Summation
I have the series
$$\sum_{k=1}^{N-1}\frac{1}{1-w_k} $$ where $w_k=e^{\frac{2\pi i k}{N}}$, how can I find the summation of this , another question related to this sum
$$\sum_{k=1}^{N-1}\frac{1}{z-w_k} ...
3
votes
4answers
77 views
Solve equation $\sqrt{4t + 1} = 3 - 3t$
Solve equation $\sqrt{4t + 1} = 3-3t$
→ I squared both sides and got ► $4t + 1 = 9 - 18t- 3t²$
→ I then moved the 3t² to the left side and combined like pairs and got ► $3t² + 12t - 8 = 0$
I'm ...
5
votes
3answers
107 views
Using the hypothesis $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ to prove something else
Assuming that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$$
Is it possible to use this fact to prove something like:
...
1
vote
2answers
56 views
Partial fraction expansion two variables
How to expand
$$\frac{y}{(x-y)(y-1)}$$
by partial fraction expansion.
3
votes
4answers
100 views
Simplify $\sqrt n+\frac {1}{\sqrt n}$ for $n=7+4\sqrt3$
If $n=7+4\sqrt3$,then what is the simplified value of
$$\sqrt n+\frac {1}{\sqrt n}$$
I was taking LCM but how to get rid of $\sqrt n$ in denominator
0
votes
1answer
38 views
Simple Math Equation find sum of 4 numbers and if greater then number X reduce all 4 numbers respectively
Im not the greatest at Math but i have the following problem:
impressions = 791.
watched 100 = 500
watched 75 = 383
watched 50 = 600
watched 25 = 700
The sum of all watched fields is 2183.
...
6
votes
2answers
74 views
Evaluate the Sum $\sum_{i=0}^\infty \frac {i^N} {4^i}$ [duplicate]
$$\displaystyle\sum_{i=0}^\infty \frac {i^N} {4^i}$$
I'm supposed to evaluate this as I'm working through Data Structures and Algorithm Analysis in C++. I've solved similar problems, and after ...
2
votes
2answers
46 views
Faulty velocity question?
If a ball is thrown vertically upward with a velocity of $160 \text{ ft/s}$, then its height after t seconds is $s = 160t − 16t^2$.
a) What is the velocity of the ball when it is $384 \text{ ...
15
votes
7answers
1k views
How to solve $x^{1/2}-x^{1/3} = 0$
How can I solve the following equation? I really can't figure out how to solve it:
$x^{1/2}-x^{1/3} = 0$
Thank you.
0
votes
1answer
16 views
Finding specific alternative form of $\frac{(x-y)x+{y\over(y-z)}}{(x+y)z}=ab$
How does one approach; $$\frac{(x-y)x+{y\over(y-z)}}{(x+y)z}=ab$$ to find the form: $$-a b z (x+y) (y-z) = x^2 (-y)+x^2 z+x y^2-x y z-y$$
1
vote
2answers
45 views
Rationalizing Denominators with Radicals
How do you rationalize this fraction:
$$\frac{1}{\sqrt{n+1}}$$
0
votes
0answers
12 views
Question involving regression line and sum of squares of residuals?
Suppose for regression line ssres=19. What is true about the sum of squares of residuals for any other linear function?
2
votes
2answers
55 views
Algebra help please
I already forget about algebra and I will be going to take an exam days from now. Please, can someone explain the steps here.. how it get the second part where it is equals to 20. I was confused ...
1
vote
1answer
23 views
Partial fraction decomposition with a nonrepeated irreducible quadratic factor
I'm trying to do a partial fraction decomposition on the following rational eqn with a nonrepeated irreducible quadratic factor:
$$\dfrac{-28x^2-92}{(x-4)^2(x^2+1)}$$
I've broken it down into an ...
0
votes
2answers
26 views
Earth Quake Question Logarithmic Type Problem
An earthquake off the coast of Vancouver Island was measured at 8.9 on the Richter Scale
and an earthquake off the coast of Alaska was measured at 6.5. How many times more intense, to the nearest ...
4
votes
1answer
63 views
Find $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ if $a+b+c=0$
I'm stuck at this algebra problem, it seems to me that's what's provided doesn't even at all.
Provided: $$a+b+c=0$$
Find the value of: ...
0
votes
1answer
34 views
How to make a unit step function?
I am trying to make a unit step function.
I have this function (the equation of an ellipse, not centered at the origin):
$$
f(x,y) = \frac{(x-X_c)^2}{a^2}+\frac{(y-Y_c)^2}{b^2}
$$
What I would ...
4
votes
1answer
40 views
Apples and their volumes
An apple has a peel that is 1cm thick and a total diameter of 12cm. What percentage of volume of the apple is the peel?
I tried
$$\frac{\text{volume}(\text{radius of 6})-\text{volume}(\text{radius ...
0
votes
2answers
25 views
Similar cones - volumes and lateral areas
Two similar cones have volumes 9$\pi$ and 72$\pi$. If the lateral area of the larger cone is 32$\pi$, what is the lateral area of the smaller cone?
I did the following...
$\frac {(9\pi)^3} ...
1
vote
3answers
51 views
Simplifying fractions - Ending up with wrong sign
I've been trying to simplify this
$$
1-\frac{1}{n+2}+\frac{1}{(n+2) (n+3)}
$$
to get it to that
$$
1-\frac{(n+3)-1}{(n+2)(n+3)}
$$
but I always end up with this
$$
1-\frac{(n+3)+1}{(n+2)(n+3)}
$$
Any ...
54
votes
11answers
3k views
Zero to the zero power - Is $0^0=1$?
Could someone provide me with good explanation of why $0^0 = 1$?
My train of thought:
$x > 0$
$0^x = 0^{x-0} = 0^x/0^0$, so
$0^0 = 0^x/0^x = ?$
Possible answers:
$0^0 * 0^x = 1 * 0^x$, so ...
0
votes
3answers
24 views
upper bound for $\frac{ax}{x-2}$
I need an upper bound for
$$\frac{ax}{x-2}$$
I know that $1\leq a< 2$ and $x\geq 0$.
This upper bound can include just $a$ and constant numbers not $x$.
thanks a lot.
1
vote
3answers
41 views
upper bound for $\frac{x+y}{ax+y}$
I need an upper bound for
$$\frac{x+y}{ax+y}$$
I know that $$1\leq a< 2$$
$x\geq 0 $ and $y\geq 0 $ . This upper bound can include just $a$ and constant numbers not $x$ or $y$.
thanks a lot.
1
vote
1answer
930 views
How to express a variable in terms of other variables in a system of equations?
I want to know how to express $a$ in terms of $x$, $y$ and $z$ in the following system (i.e., I want to find the function $a(x,y,z)$). And how to investigate (and to play with the equations) to know ...
0
votes
3answers
35 views
Partial fraction decomposition issue
I'm trying to do the partial fraction decomposition of the following rational expression:
$(x-4) / (x-2)(x-3)$
Here are the steps I preformed:
$ x-4 = A/(x-3) + B(x-2)$
$x-4 = Ax - 3A + Bx - 2A ...
2
votes
3answers
45 views
Evaluating limits involving absolute values
I have two questions regarding limits involving absolute values. How do I evaluate the following:
$\displaystyle\lim_{x\to -6} \frac{2x+12}{|x+6|}$
$ \displaystyle \lim_{x\to 6} ...
1
vote
3answers
38 views
Solving systems of equations using matrices
I'm teaching myself how to solve systems of equations using matrices and row operations. I get the basic mechanics of it (legal operations, etc.), but it seems like it's kind-of a crapshoot deciding ...
3
votes
3answers
105 views
Tricky logarithmic problem?
It is given that $\log_9 p = \log_{12} q = \log_{16} (p+q) $. Find the value of $q/p$. I can see that the bases have common factors, but I don't exactly know how to exploit that. I tried many ...
1
vote
1answer
43 views
Given $f(x) =x^2+bx+c$, $f(1) = 2$, and $f(-1) = 12$, how do I get $b$ and $c$?
Is there a different way to get $b$ and $c$ values and then the value of $f(2)$?
Here's how my book does it:
Given that
$$f(x) =x^2+bx+c, \qquad f(1) = 2,\qquad f(-1) = 12$$
we see
$$
f(1) = 2 ...
2
votes
1answer
505 views
Simultaneous equations for more unknowns than equations
I have a system of equations. This system is 8 equations for 16 unknowns. Is it possible to solve the equations for some answer? By that, I mean, it's highly likely that they will have several ...
15
votes
19answers
7k views
Proof for formula for sum of sequence $1+2+3+\ldots+n$?
Apparently $1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2$.
How? What's the proof? Or maybe it is self apparent just looking at the above? Does this problem have a name and maybe a presence on the net? ...
0
votes
1answer
24 views
Factorizing expression with exponents
I fail to see the justification that allows the textbook author to go from line 2 to line 3.
$$ C_1 e ^{u} - C_1 e^{-u} = 0$$
$$ C_1 (e^{u} - e^{-u}) = 0 $$
$$C_1 (e^{2u} - 1) = 0$$
where $u = ...
0
votes
3answers
28 views
Recurrence relation of two next terms
For the recurrence relation, $a_{n+2}=3a_{n+1}-2a_n$ with $a_0=2$ and $a_1=3$, compute the first six terms of the sequence and derive a closed form formula for this sequence.
So I'm totally lost with ...
4
votes
5answers
157 views
Checking whether a polynomial of high degree is bijective or not.
Let $P(x)$ be a polynomial of degree $101$. Then $x\mapsto P(x)$ cannot be a one-one onto mapping, i.e., bijective function from $\Bbb{R}$ to $\Bbb{R}$. True or false?
I think is when we take ...
7
votes
5answers
171 views
What does $x^\pi$ mean? [duplicate]
I was just wondering, what does $x^\pi$ or for that matter, $x$ raised to any irrational number mean? For example, I want to represent $x^2$ then that would mean $x * x$ or if I want to do ...
0
votes
1answer
14 views
Solving dependent systems
When I'm solving a system of equations and realize that I have a dependent system, I need to express the answer in terms of y = {some value} where x is any real number, OR x = {some value} where y is ...
1
vote
3answers
39 views
Finding asymptotes to general functions
I stumbled upon this problem and didn't quite understand why the solution worked. I have two questions:
Why is it that the taylor series is enough to find the oblique asymptote? Specifically, why do ...
6
votes
4answers
241 views
solving $1+\frac{1}{x} \gt 0$
In solving a larger problem, I ran into the following inequality which I must solve:
$$ 1+\frac{1}{x} \gt 0.$$
Looking at it for a while, I found that $x\gt 0$ and $x\lt -1$ are solutions. Please ...
-1
votes
1answer
89 views
How to solve this equation for $r$? [closed]
I have a problem....
I have to express unknown "r" from from this equation:
$$Y\times r=(-Z\times 2r\times L)+(z\times K^2\times 0.5L)$$
Can someone help me ?
3
votes
2answers
52 views
Log problem, $u$ substitution the only way?
Okay so basically I want to know if you can solve this log equation without the use of u substitution:
$${\log_4{\log_3{x}}} = 1$$
I believe that u substitution is the only way to solve this ...
2
votes
3answers
82 views
Solving two algebraic equations
I'm trying to prove an identity from physics. I have the following two equations ($M$ is a constant):
$$e^2 = \left(1-\frac{2M}{r}\right)\left(1+\frac{l^2}{r^2}\right)$$ and $$r = ...
