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)

2
votes
1answer
65 views

The double ${-6}$, from ${|3x + 7| = 11}$

Solve for $x$ such that $|3x + 7| = 11$. Answer. "${x = \frac {4}{3}, -6, -6}$". First rewrite the absolute value equation as two separate linear equations. In the first equation, assume that the ${3 ...
0
votes
0answers
25 views

Theorem 2 of perfect powers

I understood theorem 1 in Perfect Powers With All equal Digits but one where a is equal to $0$,{thanks to piquito and peter for helping me}. But theorem 2 seems so complicated when a is not equal to $...
-1
votes
0answers
40 views

Find a Parametrization for the following equation?

Find the parametrization for: The lower half of the parabola $y^2=x-1$ So here's what I did: I put in $$x=t$$ Then I solved for $y$ $$y^2=t-1$$ $$y=\sqrt {t-1}$$ So the parametrization would be $$...
0
votes
2answers
52 views

For what value of $c$ is $f$ periodic?

Let $f(x)=a\sin(cx)+b\cos(cx)$, where $a$, $b$ and $c$ are constants. Since $\sin$ and $\cos$ have a period of $2\pi$, if $c\in\mathbb{Z}$ then $f$ has a period of $2\pi$. How to prove the converse? ...
1
vote
4answers
54 views

What does this sentence mean? “$\lim_{x\to x_0}$ exists at every point $x_0$ in (-1,1)$.”

What does this sentence mean? $$\lim_{x\to x_0} \;\text{exists at every point}\; x_0 \; \text{in} \; (-1,1).$$ $(1,-1)$ is just an example point. The topic is finding whether limit functions ...
0
votes
1answer
30 views

Algebra question about simplying a constant from exponential

i've a question, i'm doing an exercise of differential equations, but my result is wrong due to a step that compared with wolfram alpha i don't understood. You can check the screenshot, how the $C$ ...
-2
votes
0answers
53 views

polynomial p(x) with no integer roots [duplicate]

Suppose that $p(x)$ is a polynomial with integer coefficents. $p(0)$ and $ p(1)$ are odd numbers. Show that $p(x)$ can not have integer roots. $p(0) \equiv1\mod(2)$, $p(1) \equiv 1 \mod (2) $ $p(x)=...
2
votes
2answers
30 views

Induction Sum of all odd Numbers

Show that $\sum_{k=1}^{n}(2k-1)=n^2$ Beginning: n=1 $\sum_{k=1}^{1}(2k-1)=(2*1-1)=1=1^2$ Let $\sum_{k=1}^{n}(2k-1)=n^2$ be true, then for n=p+1 $\sum_{k=1}^{p+1}(2k-1)=(p+1)^2$ has to be true too....
5
votes
3answers
73 views

Does $X^{2/2} = X^{1/1}$?

I'm having a bit of a hard time wrapping my head around how the following that I have just learned: $\sqrt{X^2} = |X|$, and I totally understand why. But, when expressed as an exponent, doesn't this ...
0
votes
4answers
59 views

Squaring a number less than zero

When we square a number, we expect the result to be much larger than the original number. But when we square a number less than zero, we get a much smaller number. Using money as an example, square 5 ...
3
votes
1answer
55 views

Find all pairs (x, y) of real numbers such that $16^{x^2+y} + 16^{x+y^2}=1$

Find all pairs (x, y) of real numbers such that $$16^{x^2+y} + 16^{x+y^2}=1$$
2
votes
3answers
1k views

Two quadratic equations with equal ratios of roots

If the ratio of roots of $ax^2+bx+c = 0\space$and $px^2+qx+r = 0\space$is same. How to find ratio of their discriminants? I don't understand this problem,what exactly is meant by ratio of the ...
2
votes
2answers
52 views

Value of $z$ in the given system of equations

If $$\{x\}+y+\lfloor{z}\rfloor=3.1$$ $$x+\lfloor{y}\rfloor+\{z\}=2.4$$ $$\lfloor{x}\rfloor+\{y\}+z=1.3$$ then find the value of $z$. My attempt: I converted fractional part of every equation to ...
1
vote
1answer
28 views

Question on integral values of a quadratic

The question is: Find the integral values of a for which $x^2-(a+10)x + 10a +1 =0 $ has integral roots. For a quadratic to have integral roots, it's D has to be a perfect square. The discriminant(...
6
votes
5answers
172 views

Calculate the limit $\lim_{x\to 0} \left(\dfrac 1{x^2}-\cot^2x\right)$

The answer of the given limit is $2/3$, but I cannot reach it. I have tried to use the L'Hospital rule, but I couldn't drive it to the end. Please give a detailed solution! $$\lim_{x\to 0} \left(\...
4
votes
1answer
42 views

Maximizing $\frac{\int_r^1xf(x)dx}{2-F(r)}$

Consider a continuous distribution on $(0,1)$ with probability distribution function $f$ and cumulative distribution function $F$. Define $$g(r)=\frac{\int_r^1xf(x)dx}{2-F(r)}$$ and let $r_M\in(0,1)$ ...
1
vote
1answer
39 views

Find $a$ given $s=\frac12at^2$ and the values of $s$ and $t$

I can't figure out how to do this type of math problem ... Not even sure what to search for on google to find proper info to learning it. sorry for the vague question.
1
vote
1answer
177 views

A problem with progressive percentage incrementation

I have absolutely no clue if any of the terms I used in the title actually exist or make sense. I'm usually good at math (relatively) but I am facing this problem today that I just cannot solve. John ...
0
votes
0answers
33 views

Find the cost of $n$ pamphlets. [on hold]

The first $100$ copies of a pamphlet cost $5$ cents each, the second hundred cost $4$ cents each, and any over $200$ cost $3$ cents each. Find the cost ($C$) of $n$ pamphlets. (Assume $n$ is more than ...
0
votes
1answer
35 views

square root of sum vs. sum of quare roots for a certain form

Given my original formula $\sqrt { \left( 1-a-b \right) \left( 1+c+d \right) } $, i notice that it is approximately equal to $\sqrt { \left( 1-a \right) \left( 1+c \right) } +\sqrt { \left( 1-b \...
0
votes
1answer
21 views

Adding and Subtracting Powers of 10

I am entirely new to this subject and am already lost. The problem is Five x Ten to the power of Negative Two, + Four x Ten to the power of Negative Three. I was told that I have to make the exponents ...
2
votes
2answers
89 views

$x^3 +y^2 +z =100z+10y+x$ What is the largest and smallest integer that satisfies this equation.

$x^3 + y^2 +z=zyx$,where $zyx$ denotes the sequence of the digits. $x^3 +y^2 +z =100z+10y+x$,where $x,y,z>0$ The maximum value of x,y,z individually can only be 9. Maximum value: $= 9^3 + 9^2 + ...
16
votes
3answers
302 views

What is $\sum_{r=0}^n \frac{(-1)^r}{\binom{n}{r}}$?

Find a closed form expression for $$\sum_{r=0}^n \dfrac{(-1)^r}{\dbinom{n}{r}}$$ where $n$ is an even positive integer. I tried using binomial identities, but since the binomial coefficient ...
1
vote
4answers
100 views

Prove that if $b^2-4ac=k^2$ then $ax^2+bx+c$ is factorizable

Prove that if $b^2-4ac=k^2$ (for some positive integer $k$ ) then $ax^2+bx+c$ is factorizable I related this to the roots of equation: $ax^2+bx+c=0$ and using roots $x_1,x_2$: $(x-x_1)(x-x_2)=0\...
0
votes
4answers
73 views

How do I find the solution(s) to my second-degree equation?

$$f(x) = x^2 - 3x$$ My attempt : $$ \begin{align} x^2-3x &= 4\\ x(x-3) &= 4\\ x-3 &= 4 \\ x &= 7\\ \end{align} $$ I managed to solve one part of this problem but that one part is ...
1
vote
2answers
29 views

Comparison of two slopes to get the best value

I'm trying to get something in excel that may be solvable by coming up with the proper formula. I believe it's a comparison of two slopes. The theory comes an economies of scale: How can I spend the ...
1
vote
3answers
49 views

Solving $\frac{x + 2}{\sqrt{1-\frac{3}{x}}}=12$

This is my attempt: $$\frac{x + 2}{\sqrt{1-\frac{3}{x}}}=12$$ $$\implies\sqrt{1-\frac{3}{x}}=\frac{x+2}{12}$$ $$\implies 1-\frac{3}{x}=\frac{x^{2}+4x+4}{144}$$ $$\implies -\frac{3}{x}=\frac{x^{2}+4x-...
4
votes
2answers
329 views

How did Sir Isaac Newton develop and formulate the famous binomial theorem?

After completing combination, I have started to read Binomial Theorem. My book only mentioned about Pascal's Triangle. And the formula was then given straightforward. But how did Sir Issac Newton ...
7
votes
4answers
207 views

Find the value of $x$ if $x^{x^4} =4$

Find the value of $x$ if $x^{x^4}=4$. Given options are $2^{1/2}$ $-2^{1/2} $ Both 1. & 2 None of the above From option verification, we get option 3. as correct one. But is there any real ...
0
votes
0answers
68 views

Proof of Ramanujan's famous cubic identity

Ramanujan found that given a polynomial $y=x^3+ax^2+bx+c$, one can find $\sqrt[3]{u+x_1}+\sqrt[3]{u+x_2}+\sqrt[3]{u+x_3}=\sqrt[3]{3\sqrt[6]{d}+w}$ where $$d=\frac {4(a^2-3b)^3-(2a^3-9ab+27c)^2}{27}$$$$...
3
votes
3answers
49 views

construct polynomial from other polynomials

If I have a polynomial, P, with root $a$ and a polynomial, Q, with root $b$, is there a way to construct polynomial R such that $a+b$ is a root of R? Here's a concrete example. a = $\sqrt2$. $P(x) = ...
1
vote
2answers
24 views

Choosing right argument expansion for trigonometric equation

Given equation $$\cos(x) + \cos(2x) + \cos(3x) + \cos(4x) = 0,$$ which argument expansion will be the most convenient?
-1
votes
4answers
58 views

Isolate y in this equation: $\frac{y+1}{y-1} = e^{-2x} + C$

I need to isolate $y$ in: $$\frac{y+1}{y-1} = e^{-2x} + C$$ where $C$ is a constant This is a simple thing that resulted from me trying to solve $\frac{d}{dx} = (y-1) (y+1)$, but has me stumped. ...
37
votes
25answers
34k 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? PS: This problem is know as "The sum of the first $n$ positive ...
2
votes
2answers
67 views

How much velocity can a canister of fuel give a spaceship?

I've recently considered the issue of how much velocity a canister of fuel can provide a 'spaceship'. I assumed we could approximate a basic solution If we know the mass of the fuel $m$, the mass of ...
2
votes
6answers
92 views

Prove by induction $3+3 \cdot 5+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$

My question is: Prove by induction that $$3+3 \cdot 5+ 3 \cdot 5^2+ \cdots +3 \cdot 5^n = \frac{3(5^{n+1} -1)}{4}$$ whenever $n$ is a nonnegative integer. I'm stuck at the basis step. If I ...
6
votes
2answers
522 views

$\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-…}}}} $, what is the nearest value of $y^2 - y$?

I found this question somewhere and have been unable to solve it. It is a modification of a very common algebra question. $\text{Let }y=\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5-...}}}} $, what is the ...
-1
votes
1answer
44 views

separate $a$ & $b$ in $ssrt(a^a*b^b)$

It is already known that $ssrt(a^a*b^b)$ does not equal $ssrt(a^a)*ssrt(b^b) = a*b$ Is there any other method to separate $a$ and $b$? ****Please note that $ssrt$ is "super square root". and my ...
1
vote
2answers
37 views

Compound propositions as assertions?

According to comments on my previous question, compound propositions are not assertions; i.e. the statement "$p \vee q$" does not mean "$p$ (is true) or $q$ (is true)", and it does not mean "$(p$ or $...
2
votes
2answers
47 views

The maximum value of expression $ \sqrt{\sin^2x+ 2a^2} - \sqrt{-1 -\cos^2x+ 2a^2} $

If $a,x\in\Bbb R$, what is the maximum value of the expression $ \sqrt{\sin^2x+ 2a^2} - \sqrt{-1 -\cos^2x+ 2a^2} $? I tried to differentiate but it became messy.
-1
votes
3answers
66 views

Sketch the graph of a function

Hello I'm currently teaching myself precalc over the summer to get ahead in my school so I'll be on this website often to aid me in questions I may have regarding the class. I need to know how to ...
-3
votes
0answers
23 views

Sketch the graph of a function that satisfies the following requirements [duplicate]

Hello I'm currently teaching myself precalc over the summer to get ahead in my school so I'll be on this website often to aid me in questions I may have regarding the class. I need to know how to ...
0
votes
2answers
44 views

proving theorem about perfect powers

Im currently studying the journal entitled Perfect Powers with All Equal Digits but One theorem: For a fixed integer $l \geq 3$, there are only finitely many perfect $l$-th powers all whose digits ...
1
vote
2answers
37 views

Is it possible to find the partial sum of a series if I know the infinite sum?

Say, I know the value of $\displaystyle\sum_{r=0}^{\infty}t_r$ (assuming it is convergent) and I want to find out the $n$th partial sum $\displaystyle\sum_{r=0}^{n}t_r$ for the sequence $\{t_r\}$ from ...
2
votes
3answers
66 views

For how many 3-digit prime numbers $\overline{abc}$ do we have: $b^2-4ac=9$?

For how many 3-digit prime numbers $\overline{abc}$ do we have: $b^2-4ac=9$? The only analysis I did is: $(b-3)(b+3)=4ac \implies\ b\geq3 $ $b=3\implies\ c=0\implies impossible!!$ So I deduced that $...
-1
votes
2answers
48 views

Solve three equations for three unknowns. [duplicate]

So I have the following three equations which I do not know how to solve: -D * x - E * y = A + (R * D) E * F * x - D * F * y - G * z = B - (R * E * F) E * G * x - D * G * y + F * z = C - (R * E * G)...
-1
votes
2answers
45 views

$((-a)^3)^{1/2}\ne(-a)^{3*1/2}?$

The original problem is $\sqrt{(-a)^{3}}\sqrt{(-a)}$ I attempted to solve this as the following way: $\sqrt{(-a)^{3}}\sqrt{(-a)} = (-a)^{\frac{3}{2}}(-a)^{\frac{1}{2}}=(-a)^{2}=a^{2}$ However, I ...
2
votes
4answers
54 views

Positive integers $a$ and $b$ are such that $a+b=a/b + b/a$. Find $a^2+ b^2$.

Positive integers $a$ and $b$ are such that $$a+b=a/b + b/a$$ Find $$a^2+ b^2$$ My try:- Given that $$a+b=a/b + b/a$$On simplification we get $$a^2 b+ b^2 a= a^2 + b^2$$ But in my book the given ...
0
votes
1answer
13 views

Convert from one format to other

Sorry for my lack of knowledge but I need help with, it seems, a basic math algebra. I want to know how they got this: $(1 - t)P1 + tP2$ from this: $P1 + t(P2 - P1)$ I did the math and they are ...
2
votes
2answers
71 views