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)

1
vote
5answers
37 views

How to write absolute value as a “true” function

Here's the basic absolute value ... what? \begin{align*} |x| = \left\{ \begin{array}{r@{\quad \mathrm{if} \quad}l} x & x > 0, \\ 0 & x = 0, \\ \!\! -x & x < 0. \end{array} ...
0
votes
2answers
17 views

Factor Theorem (Finding values of a and b)

Question: The polynomial $p(x) = 2x^3 - ax^2 + bx + 48$ has $(x-4)$ as a repeated factor, find the values of $a$ and $b$. What I have attempted if $x-4$ is a factor then $x = 4$ is a ...
0
votes
2answers
30 views

solving system of equations involving imaginary numbers

What are the values of $a,b,c$ given the system of equations given below: $a+b+ab=i$ $b+c+bc=2i$ $c+a+ac=3i$
0
votes
0answers
27 views

What is my special quadratic?

Start with $f(x)=x^2+bx+c$. Then, attempt to solve for $x$ in $f(x)=x$. It is easily found that $x=\frac{1-b\pm\sqrt{(b-1)^2-4c}}{2}$. Then, start again with $f(x)=x$ and apply the function $f$ to ...
2
votes
0answers
14 views

Is this a valid way for performing polynomial division?

While attempting to divide a quartic by a quadratic factor to find the other factors of the given quartic, I can't help feeling I "invented" a way of dividing polynomials. Suppose you have a quartic ...
-3
votes
0answers
24 views

How do I get these values? [on hold]

I want to know how to get these answers: $A(12836.3)=42227.7$ $A=3.28971$ (phase) $\theta+46.7364=0$ $\theta=-46.7364$ from this equation. I am confuse is there a formula or a trig. identity? ...
3
votes
0answers
26 views

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$).

Find all real $x$ such that $1990[x] +1989[-x]=1$ (where $[x]$ is the floor function for $x$). My effort Rearranging our equation we have : \begin{array}{c} 1990[x]+1989[-x]&=1 \\ ...
1
vote
1answer
30 views

Struggling with BIDMAS.

This question came up an I'm not sure about it: You need to simplify leaving the answer in standard form: $\dfrac{2((3-3^2)^2)}{3+\sqrt{4^2-7}}$ I struggle to work it out myself. When I used a ...
6
votes
4answers
103 views

Solve equation $\frac{1}{x}+\frac{1}{y}=\frac{2}{101}$ in naturals

My try was $$\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}=\frac{2}{101}\\x+y=2k,xy=101k\\x=2k-y\\y(2k-y)=101k\\2ky-y^2=101k\\y^2-2ky+101k=0\\y=k+\sqrt{k^2-101k}\\x=k-\sqrt{k^2-101k}$$ Now $\sqrt{k^2-101k}$ ...
1
vote
1answer
53 views

Why do you need absolute value when taking $\sqrt{\cos^2(x)}$

$$\sqrt{\cos^2(x)} = |\cos(x)|$$ Is this on the right track? If you have an underlying $\cos(x)$ that is negative, and then you square it, you will now have $\cos^2{x}$, which is positive. But, if ...
4
votes
2answers
47 views

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$?

Is there an identity that says $|\sqrt {a^2+x^2} - \sqrt {a^2+y^2}| \leq |\sqrt {x^2} - \sqrt {y^2}|$? Because of the nature of the square root function, its derivative monotonically decreases. so ...
0
votes
2answers
37 views

What is my mistake

Spot my mistake: $$\frac{\left(\text{P}_1+\text{P}_2+\dots+\text{P}_n\right)-\left(\text{Z}_1+\text{Z}_2+\dots+\text{Z}_n\right)}{n-m}\le-\ln(50)$$ ...
0
votes
2answers
22 views

Simplifying Expression Factorial Expression

I'm confused as how I'm meant to simplify this:$$\frac{(n-2)!}{(n-2-r)!}$$ I have other factorial questions where the variable isn't present in the top factorial like the question above and I'm ...
7
votes
5answers
65 views

Proving that $\cos(\frac{\arctan(\frac{11}{2})}{3}) = \frac{2}{\sqrt{5}}$

I am trying to solve the cubic equation $x^3-15x-4=0$ using Cardano's formula. I already know that the solutions are $x=4$, $x= \sqrt{3}-2$ and $x= -\sqrt{3}-2$ and that using the formula in this ...
1
vote
2answers
38 views

$N + 2N + 3N +\dotsb$ series?

I've been trying to figure this out but no luck yet. What is the general formula for the sum $N + 2N + 3N + 4N +\dotsb$? Could it be equal to something like $$N + 2N +3N + ..... + xN = ...
1
vote
0answers
22 views

Difference Equations and Discrete Integral and Derivative

So I'm trying to learn difference equations, and the book that I'm using defines the following: The discrete derivative of a function $a_n$ of the integers is defined as: $$ D a_n = a_{n+1} - a_n $$ ...
0
votes
4answers
74 views

Is this argument valid for a proof?

Please kindly forgive me if my question is too naive, i'm just a prospective undergraduate who is simply and deeply fascinated by the world of numbers. My question is: Suppose we want to prove that ...
0
votes
2answers
49 views

Need help solving an equation: $f(x)=\frac{1}{x}+\ln{x}-2=0$ [on hold]

How would I solve this equation: $$f(x)=\frac{1}{x}+\ln{x}-2=0$$ What I noticed was that $\frac{d}{dx}\ln{x}=\frac{1}{x}$. Don't know how this would help.
0
votes
2answers
50 views

Solving algebraically a cubic inequality

Is there any way to solve algebraically the following inequality: $x-x^{3}+x^{2}-5 \geq 0$. I know the answer is $x \leq -1.594$ because I have plotted the function but I can't figure out how to do ...
3
votes
3answers
67 views

What's the equation for a rectircle? (Perfect rounded-corner rectangle without stretching on only one dim)

The equation for a rounded square seems to be: $x^4 + y^4 = 1$ You can make the radii smaller by increasing (over the even integers) the exponents in the equation. Here's a picture: Wolfram Alpha ...
2
votes
0answers
33 views

A subset of easily solved 4th degree polynomials

I've found (maybe, maybe not, but it's not on this Wikipedia or this Wikipedia) that there is a subset of easily solved quartic polynomials of the form ...
0
votes
4answers
56 views

number of real solution of $\sin x+2\sin 2x = 3+3\sin 3x\;\forall x\in \left[0,\pi\right]$

Number of real solution of $\sin x+2\sin 2x = 3+3\sin 3x\;\forall x\in \left[0,\pi\right]$ My try: I have tried graphing. Using the graphs of $$y=\sin x+2\sin 2x \;\;\; \text{and} \;\;\; ...
0
votes
3answers
36 views

Show that $x^3y^3(x^3+y^3) \leq 2$

If $x,y$ are positive reals such that $x+y = 2$ show that $x^3y^3(x^3+y^3) \leq 2$. I thought about working backwards on this one. We can try to prove that $x^3y^3(x^3+y^3) \leq 2$ is true by ...
1
vote
1answer
18 views

Distance between a point $X$ and a line $2x-y = 1$

I am asked to state a condition for a point $X$ such that the point $X$ is a distance of 10 units from the line $2x-y = 1$, and then find the Cartesian equation of the set of all points $X$ that are a ...
3
votes
3answers
83 views

Solve the equation $7t+[2t] =52 $ ,where $[x]$ denotes the floor function for $x$.

Solve the equation $7t+\left\lfloor 2t\right\rfloor =52 $. My effort Using the fact that for any number $x$ we have that $x=\left\lfloor x\right\rfloor+\{x\}$ (where $\{x\}$ is the fractional ...
0
votes
2answers
13 views

Comparing the asymptotic growth of certain functions.

Consider some nonconstant functions $f(x)$ and $g(x)$, and suppose $\lim_ {x \to a} f(x) > \lim_{x \to a} g(x)$ for some nonzero $a$. Can we somehow conclude that $f(x) - g(x)$ is of constant sign ...
-3
votes
0answers
24 views

mixing of three type of teas [on hold]

Three tea, whose prices per kg are respectively 15dollars,25dollars and $30, are to be taken two at a time and mixed in the same proportion so that the resulting mixtures are of equal value. How may ...
-1
votes
2answers
109 views

What does it mean to say “again” or “finally” in math? [on hold]

According to math rules if we say again, does it mean we are saying repeat previous step? For example: You have $10$ coins. Add $10$ more coins. Add $2$ coins Again $2$. Again $2$. Add $2$. And ...
1
vote
2answers
66 views

If $ \frac{x}{b-c} = \frac{y}{c-a} = \frac{z}{a-b}$, Prove that $x+y+z$ $=0$

Question If $$ \frac{x}{b-c} = \frac{y}{c-a} = \frac{z}{a-b} $$ Prove that $x+y+z$ $=0$ I've attmempted this question by cross multiplying so that $$ x(c-a)(a-b) = y(b-c)(a-b) = ...
0
votes
1answer
43 views

Finding a sixth degree polynomial that goes through 8 points

For a summative math research assignment, I will have to find a sixth degree polynomial that would ideally go through the following points: (0, 20.5625) (10, 27.5625) (30, 14.5625) (50, 14.6875) (60, ...
2
votes
2answers
53 views

How do I deal with a floor function is a system of equations?

How would one solve an equation with a floor function in it: \begin{cases} y=12(x-\lfloor x \rfloor) \\ x=12(y-\lfloor y \rfloor) \end{cases} Maybe an algebraic method could be used?
0
votes
1answer
46 views

Do there exist $a,b,c,d,e,f$ such that $ax^2+by^2+cxy+dx+ey+f > 0 \quad\forall 0<x\le 1, 0< y\le 1$ and…

Do there exist $a,b,c,d,e,f$ satisfying: \begin{cases} ax^2+by^2+cxy+dx+ey+f > 0 \quad\forall 0<x\le 1, 0< y\le 1\\ a+b+c+d+e+f \le 1\\ a+d+f \le 0\\ b+e+f \le 0\\ f\le 0 \end{cases}? ...
1
vote
2answers
50 views

Find the smallest positive value taken by $a^3+b^3+c^3-3abc$

Find the smallest positive value taken by $a^3+b^3+c^3-3abc$ for positive integers $a,b,c$. Find all integers $a,b,c$ which give the smallest value. Since it is generally hard to find the minimum ...
0
votes
0answers
45 views

Finding the symmetries of $f(x)$ given that $f(f(x))=x$

If a function $f$ satisfies the property $f(f(x))=x$, then how would you show that it is an odd function? I tried the following but couldn' t get anywhere. Also, would it have any other symmetries? ...
0
votes
1answer
36 views

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$

$(X_1+X_2+ X_3 + \cdots + X_n)^2 =$ $?$ with $X_i$'s $ \in \mathbb{R}$ Just from computing $(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc$ I am guessing the general formula is: $(x_1 + \cdots + ...
2
votes
1answer
42 views

How can I find an output of this function's inverse without graphing?

How can I find $f^{-1}(5)$ where $$f(x)=\frac{27}{\pi}x + \sin x$$ algebraically? Thank you!!
0
votes
2answers
38 views

Is there an easier way to solve big systems of equations?

I have the system \begin{equation*} \begin{cases} 4x^2 - 3xy + 9y^2 = 15,\\ 2x + 3y = 5 \end{cases} \end{equation*} Is there any better way than to substitute $\frac{5-2x}{3}$ in for $y$?
1
vote
1answer
20 views

Convert base y to base 10

I have a problem to find base y. The equation given is ($1111011$)gray + ($123$)y + ($211.1$)3 + ($34.4$)$6$ = (CD)$16$ + ($40$)y - ($10010$)BCD. I am able to simplify by converting everything ...
4
votes
2answers
71 views

$\frac{1\cdot2^2+2\cdot3^2+\cdots+n(n+1)^2}{1^2\cdot2+2^2\cdot3+\cdots+n^2(n+1)}=\frac{3n+5}{3n+1}$ by Mathematical Induction

Prove by Mathematical Induction: $$\frac{1\cdot2^2+2\cdot3^2+\cdots+n(n+1)^2}{1^2\cdot2+2^2\cdot3+\cdots+n^2(n+1)}=\frac{3n+5}{3n+1}$$ Now by inductive hypothesis: ...
0
votes
1answer
9 views

Understanding the steps taken in a calculation of the maximum profile likelihood of a simple ODE, given some data

I'm trying to understand a calculation made in a paper (section 2 from the supplementary contents of ...
0
votes
2answers
20 views

Matrix Solving Method

Solve my matrix method: $$3y+4x=2xy$$ and $$9y-2x=\frac{5xy}{2}$$ My solution Here: $$3y+4x=2xy$$ Dividing both sides by $xy$ $$\frac{3}{x}+\frac{4}{y}=2$$----(1) Again, $$9y-2x=\frac{5xy}{2}$$ ...
1
vote
1answer
22 views

cauchy- schwarz inequality b/2a input value

I was watching this video but at 8:05 I don't get why to solve for the function $p(t) = at^2 + bt + c \geq 0$, Sal decides to input $t= \frac{b}{2a}$. Someone made this explanation: $\frac{b}{2a}$ is ...
0
votes
1answer
51 views

Find the two square roots of a complex without transforming to the trigonometric form. [on hold]

How to solve this? I know it will be solved with De Moivre's Theorem but i don't know how. Find the two square roots of $\frac{-7+3i}{-3-7i}+\frac{-3-7i}{7-3i}$ without transforming to the ...
2
votes
1answer
50 views

Positive integers $(x,y)$ their sum and product is a perfect square

Is there any genuine approach to find pair of positive integers $(x,y)$ such that both their sum and product is a perfect square? One pair is $(5,20)$ but it looks to me that this question can be ...
0
votes
1answer
24 views

how to write as geometric series $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ [on hold]

How would I write $\dfrac{A(3s-5)}{(s-3)(3s-5)}+\dfrac{B(s-3)}{(3s-5)(s-3)}$ as a sum of geometric series?
1
vote
2answers
39 views

Can $(\mu_1 - \mu)^2 + (\mu_2 - \mu)^2$ be simplified to remove $\mu?$

$\mu_1$ and $\mu_2$ are means of two groups, while $\mu$ is the overall mean. I feel like this can be done with some basic algebra I've forgotten, but I'm not sure. Here is some more context: $$ f ...
0
votes
2answers
31 views

Q: Quadratic Division - How to divide two quadratics?

My studies into graphs and models following examples from Khan Academy has helped me on my goal to learn how to chart and model via the quadratic formula However while I have been successful in ...
2
votes
2answers
47 views

Is $f(x) =x^{-1}$ an analytic function?

As a prospective undergraduate who is doing pre-study in preparation for my future endeavours, i recently learnt about analytic functions and would like to know whether $f(x) = x^{-1}$ is analytic in ...
-1
votes
1answer
36 views

Is my proof by induction correct?

If $x_1 , x_2,......x_n$ are non-zero elements of a field so is $\prod_{k=1}^n x_k$; and $\left(\prod_{k=1}^n x_k\right)^{-1} = \prod_{k=1}^n x_k^{-1}$. Assume $n = 2$ true; How I did it: First: ...
1
vote
2answers
26 views

Why coefficients have to be proportional for two quadratic functions to have the same roots?

We have the next two quadratic functions: $ ax^2 + bx + c = 0 $ $ mx^2 + nx + p = 0 $ If $ a/m = b/n = c/p $ then they have the same roots. What is the intuition behind this statement?