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.

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3
votes
2answers
97 views

prove that there is some number $x$ such that $x^{179}+\frac{163}{1+x^2+\sin^2x}=119$ [closed]

prove that there is some number x such that $ x^{179}+\frac{163}{1+x^2+\sin^2x}=119$ I really want to know the general process to prove something like that...
1
vote
1answer
76 views

Solve an equation with a finite chain of nested radicals

Solving an infinitely long square root problem is easy but how to solve this one? The equation goes like this. $$\sqrt{4+\sqrt{4+\sqrt{4-\sqrt{4-x}}}} = x$$
1
vote
1answer
43 views

Inequaties with modulus on both sides [duplicate]

I need help with inequalities with modulus on both sides. For instances: |x| ≤ |x − 2| I've been doing: ...
1
vote
1answer
83 views

Does $4^k = 4 \times 4^k$?

I am trying to separate $4^k$.. I have tried breaking it up as both as $4\times4^k$ or $2 \times 2^k$.. But both of these feel wrong. I have reviewed the exponent laws and know how to use them with ...
2
votes
3answers
56 views

Primitive roots: If $p$ is a prime such that $p\equiv 1 \pmod 4$, and $a$ is a primitive root, then $-a$ is also a primitive root. [duplicate]

Primitive roots: If $p$ is a prime such that $p\equiv 1 \pmod 4$, and $a$ is a primitive root, then $-a$ is also a primitive root. In this particular question I did show that in fact $(-a)^{p-1} ...
1
vote
5answers
74 views

roots of the quadratic equation $(a^4+b^4)x^2+4abcdx+(c^4+d^4)=0$ are real

If $a,b,c,d\in \mathbb{R}$ and roots of the quadratic equation $(a^4+b^4)x^2+4abcdx+(c^4+d^4)=0$ are real.Then prove that roots are equal. $\bf{My\; Try::}$ Given ...
2
votes
2answers
25 views

Equation with absolute value

I tried to solve my first equation with absolute value: $$ |x+1 | < 0.01 $$ if $ x < (-1) $ then $x+1$ is negative, so $|x+1| = -(x+1)$ : $ -x-1 < 0.01 $ $ -x < 0.01 + 1 $ $ -x < ...
1
vote
1answer
27 views

The diagonal product of the family $\{f_\alpha:\alpha\in I\}$ is injective.

Suppose that $f_\alpha : X\longrightarrow Y_\alpha$ for any $\alpha\in I$, where $X$,$Y_\alpha$ be nonempty sets for all $\alpha\in I$, and Let $f : X\longrightarrow \prod\{Y_\alpha: \alpha\in I \}$ ...
3
votes
3answers
106 views

How to find $abc$ if one is given the values of $a + b + c,$ $a^2 +b^2+c^2, \ a^3+b^3+c^3$ and $ac+bc+ab,$

If I am given the values of $a + b + c,$ $a^2 +b^2+c^2, \ a^3+b^3+c^3$ and $ac+bc+ab,$ how do I find value of $abc \ ?$ I expanded $(a+b+c)^3$ to get $a^3+b^3+c^3 + ...
0
votes
1answer
42 views

How do I prove this claim using the contrapositive?

For all positive integers $x$, if $x^2-4x+1$ is even, then $x$ is odd. Steps I took: So, I know that the contrapositive of $(a \Rightarrow b)$ is $(\neg b \Rightarrow \neg a)$ Proof: We prove the ...
0
votes
2answers
46 views

How to understand this algebraic contradiction and relate to defintion of complex numbers?

Using the following identities: $x^b \cdot x^a = x^{b+a}$ (#1) Example: $100^1 \cdot 100^{-1} = 100^{1-1} = 1 $ $(yz)^c =y^c \cdot z^c $ (#2) Example: $16^{3/2} \cdot \sqrt{9}=16^1 \cdot ...
3
votes
1answer
62 views

prove that $\sum_{k=1}^{n=90}\frac{1}{\sin(k-1)\sin(k)} =\frac{\cos1^{\circ}}{\sin^21^{\circ}}$ [duplicate]

how do I prove that $$\sum_{k=1}^{n=90}\frac{1}{\sin(k-1)\sin(k)} =\frac{\cos1^{\circ}}{\sin^21^{\circ}}$$ or $${1\over \sin1^{\circ}\sin2^{\circ}}+{1\over \sin2^{\circ}\sin3^{\circ}}+{1\over ...
3
votes
2answers
67 views

How do I find the minimal polynomial for 11th roots of unity sum $\zeta+\zeta^{-1}$. Non-trivial seemingly

Let $\zeta$ be a primitive 11th root of unity, how does one show that $\alpha= \zeta+\zeta^{-1}$ has minimal polynomial $x^5+x^4-4x^3-3x^2+3x+1$? I have been unable to solve this problem that I would ...
0
votes
1answer
34 views

$\sum_{k=1}^{n}{T_k}$ $ = {1\over 2}{(2n+1)^{3/2}-1}$

if $$T_n = {4n+\sqrt{4n^2-1}\over \sqrt{2n+1}+\sqrt{2n-1}}$$ (a).prove that $$\sum_{k=1}^{n}{T_k} = {1\over 2}\{{(2n+1)^{3\over2}-1}\}$$ And (b)use telescopic series to prove the result.
2
votes
1answer
38 views

prove $n!(n+1)^2-1$ using telescoping series

the general formula of $$\sum_{k=1}^{n} (k!)(k^2+k+1) $$ I got it using telescoping series. in the form of $n$. $$n!(n+1)^2-1$$ How do i prove the general formula using Telescoping series?
1
vote
3answers
67 views

if $a(b-c)x^2+b(c-a)x+c(a-b)=0$ has repeated roots prove…

if the equation $$a(b-c)x^2+b(c-a)x+c(a-b)=0$$ has repeated roots prove that $${1\over a}, {1\over b},{1\over c} $$ are in Arithmetic Progression Any idea about how to go about solving this ? Thanks ...
31
votes
8answers
3k views

Can a pre-calculus student prove this?

a and b are rational numbers satisfying the equation $a^3 + 4a^2b = 4a^2 + b^4$ Prove $\sqrt a - 1$ is a rational square So I saw this posted online somewhere, and I kind of understand what ...
79
votes
14answers
7k views

Can I think of Algebra like this?

This year in Algebra we first got introduced to the concept of equations with variables. Our teacher is doing a great job of teaching us how to do them, except for one thing: He isn't telling us what ...
2
votes
3answers
46 views

Numerical Identities

Please can someone explain if this identity is correct: |a| = $\sqrt{a^2} \ $ I thought it should be: |a| = $(\sqrt{a})^2\ $ being that the former would produce an answer that is either positive or ...
1
vote
1answer
40 views

How to prove algebraic theorem of the solution set of the equation ax+b=c using field axioms

Prove the following Theorem: If a, b, c are numbers, the solution set of the equation ax + b = c consists of either (a) a single number, (b) the empty set, or (c) the entire real line. Hint: If you ...
-3
votes
1answer
50 views

Exponential Proof

Let $c(x)=\dfrac{3^x+3^{-x}}{2}$ and $s(x)=\dfrac{3^x-3^{-x}}{2}$. Show that $(c(x))^2=\frac{1}{2}(c(2x)+1)$. How does one go about solving this? I have honesty tried substituting in ...
1
vote
1answer
86 views

Can someone solve this equation?

I am trying to solve the following equation. $$ \frac{d}{dx}\left[\frac{b}{x}\left({n+2^x}\right)\right]=0 $$ My trial: \begin{align} \frac{d}{dx}\left[\frac{b}{x}\left({n+2^x}\right)\right] ...
0
votes
0answers
22 views

Clarification regarding : How to find the maximum value of $3^x+5^x−9^x+15^x−25^x$ as x varies over the reals? [duplicate]

How to find the maximum value of $3^x + 5^x - 9^x + 15^x - 25^x$ as x varies over the reals? In the above post made by me, @JVV wrote an answer using the method of partial derivatives and putting ...
1
vote
2answers
59 views

The sum of non real roots of the polynomial equation $x^3+3x^2+3x+3=0$

Problem : The sum of non real roots of the polynomial equation $x^3+3x^2+3x+3=0$ (a) equals 0 (b) lies between 0 and 1 (c)lies between -1 and 0 (d) has absolute value bigger than 1 My ...
0
votes
0answers
57 views

Factoring a quadratic polynomial, $4T^{2}-48T+144$

The question is asking me to factor the following polynomial to the simplest form. (without making it messy) \begin{align*} & 4T^{2}-48T+144\\ \end{align*} Here is how I do it but not sure which ...
2
votes
1answer
35 views

Algebraic word problem involving auxiliary variable

When Mr. and Mrs. Smith took the airplane, they had together 94 pounds of baggage. He paid 1.50 and she paid 2.00 for excess weight. If Mr. Smith made the trip by himself with the combined baggage of ...
0
votes
1answer
60 views

Algebraic word problem [closed]

A woman walked five hours (total), first along a level road, then up a hill, then she turned around and walked back to her starting point along the same route. She walks 4 miles per hour on the level, ...
7
votes
2answers
146 views

Prove that $\sqrt[3]5 - \sqrt[4]3$ is Irrational

I've gone many directions and they all fail. The sum of two irrationals doesn't need to be irrational. I found a proof saying: if irrational $x,y$ have a rational sum $x+y$, then $x-y$ is ...
1
vote
2answers
35 views

Number of zeroes in a particular interval [-1,1] for $x^{2n+1} + (2n + 1) x + a = 0$

Let n be a natural number and let a be a real number. The number of zeros of $x^{2n+1} + (2n + 1) x + a = 0$ in the interval $[-1, 1]$ is ? ...
3
votes
4answers
153 views

How to find the maximum value of $3^x + 5^x - 9^x + 15^x - 25^x$ as x varies over the reals?

How to find the maximum value of $3^x + 5^x - 9^x + 15^x - 25^x$ as x varies over the reals ? Suggestions please!
4
votes
5answers
60 views

In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions?

In the real number system,the equation $\sqrt{x+3-4\sqrt{x-1}}+\sqrt{x+8-6\sqrt{x-1}}=1$ has how many solutions? I tried shifting the second term to the rhs and squaring.Even after that i'm left with ...
0
votes
1answer
17 views

Solve equation when function is min

I have the following function $$ Q=F(L,K) = min (L,K) , L,K>0 $$ And I would like to solve the equation for K. So as to make the graph L,K Could you help please ?
1
vote
1answer
74 views

Strange problem with the imaginary unit [duplicate]

In class while messing with fractions and complex numbers I found this "paradox" $$ \sqrt{-1}=\sqrt{-1} $$ $$ \sqrt{\frac{-1}{1}}=\sqrt{\frac{1}{-1}} $$ $$ ...
-1
votes
1answer
42 views

Determine the symmetric sum of roots.

Please no complete solutions, ONLY HINTS REQUESTED! The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the ...
3
votes
5answers
107 views

Why is $x-(x-1)=1$ instead of $x-x-1=-1$ as in $a*(b*c)=a*b*c$?

As in the title of the question, why is $x-(x-1)=1$ instead of $x-x-1=-1$ as in $a*(b*c)=a*b*c$? Is there any intuitive example to explain it?
0
votes
1answer
10 views

checking of removable discontinuity for sinusoidal function

before i will post itself question,let us consider following definition now consider following question and part $c$ related to part $c$, we have in square brackets $sin(x)$ for $sin(x)$ ...
1
vote
3answers
157 views

The number of distinct real roots of a polynomial

I have trying to solve this problem for a long time now. After having read related concepts, I am still stuck. The problem is as follows- Find the number of distinct real roots of the equation ...
0
votes
1answer
21 views

When is an equation independent of a certain variable?

Consider an algebraic equation, whose sides are constituted only by products of functions of single variables. $$f(x)g(y)h(z) = i(x)j(y)k(z)$$ $f,g,h,i,j,k$ are complex functions of real variable ...
1
vote
1answer
20 views

Show difference between values of turning points of $ f(x) = (c-\frac{1}{c}-x)(4-3x^2) $

How would you show that the difference between the values of the turning points of $$ f(x) = (c-\frac{1}{c}-x)(4-3x^2) $$ is $$ \frac{4}{9}(c+\frac{1}{c})^3 $$ $c>0$ I have attempted to ...
-5
votes
2answers
52 views

How to solve $3 (2x-3) + 5 (3x-14) = 14$ [closed]

I just want to ask that how to solve this Linear Equation $$3 (2x-3) + 5 (3x-14) = 14$$
0
votes
1answer
39 views

Limit of quotient of factorial and ceiling

I am trying to study some calculus but I am faced with a problem. I need to compute $$\lim_{x \rightarrow \infty} \frac{x!}{(\lceil x \rceil)}$$ I have tried to use $$ \lim_{x \rightarrow \infty} ...
0
votes
2answers
68 views

How to find the composition of case-defined functions?

Let $$g(x)= \begin{cases} 3+x & \text{if $x\leq0$} \\ 3-x &\text{if $x > 0$} \end{cases}$$ Find $f$ if $f$ is defined as $f(x) = g(g(x))$. How to solve the problem ...
1
vote
1answer
31 views

A quadratic polynomial $f$ such that $f\circ f' = f'\circ f$

Given that $\ f\left( x \right)=ax^{2}+bx+c$, find a value for each of $a, b$ and $c$ such that: $f\left( f'\left( x \right) \right)=f'\left( f\left( x \right) \right)$. What I did: ...
1
vote
1answer
26 views

Greatest value of modulus of z

Given that the equation $z^2=3+4i$ How to find the greatest value of modulus of $z$? And how to find the difference between the largest and the least values of art $z$? I found the z is equal to ...
0
votes
1answer
34 views

$C_1+2.5C_2 +3.5^2C_3 +\cdots n.5^{n-1}C_n = ? $

Problem : $C_1+2.5C_2 +3.5^2C_3 +\cdots n.5^{n-1}C_n = ? $ My approach : $C_1+2.5C_2 +3.5^2C_3 +\cdots C_n = \sum_{n=1}^n n.5^n C_n$ Also $C_0+C_1+C_2+\cdots C_n = 2^n$ Please suggest how to ...
3
votes
3answers
70 views

Looking to confirm inequality or learn where mistake is

Hello I am looking for some advice on the following, I am wanting to show that $n^{\frac{1}{n}} \lt (1+\frac{1}{\sqrt{n}})^{2}$ for all $n \in \mathbb{N}$ and I thought I would try by induction ...
2
votes
3answers
114 views

Making radical simplications like $\frac{\sqrt{60}}{6} = \sqrt{\frac{5}{3}}$

I discovered that $\frac{\sqrt{60}}{6} = \sqrt{\frac{5}{3}}$. Is there a way of generally doing this? How?
-1
votes
2answers
59 views

Equation of parabola with given $y$-intercept and roots

I have two questions involving quadratics. Next to a diagram of a parabola with a ma point with the y intercept of $(0,p)$ and roots $(-1,0)$ and $(p,0)$ it says: 3a) Show that the equation of the ...
1
vote
2answers
75 views

Find all values of $x$

Determine all real values of $x$ such that: $$\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x) $$ Let $u = 2^x$ and let $y = 3^x$ For ease, let $\log_{2}$ be represented by just $\log$ so: ...
3
votes
2answers
52 views

Find $s^4-18s^2-8s$

Let $a,b,c$ be the roots of $x^3-9x^2+11x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^4-18s^2-8s$. $s^4 - 18s^2 - 8s = (s)(s + 4)(s - 2 + \sqrt{6})(s - 2 - \sqrt{6})$ $P(x) = (x - a)(x ...