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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0
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2answers
46 views

For what values of $a,b,c$ will $ax^2+bx+c \geq 0$ hold $\forall x \in \mathbb{R}$?

If I let $y=ax^2+bx+c, (a\neq 0)$ then extremum of $y$ is attained at $x=-\frac{b}{2a}$. Then $\large\frac{\mathrm {d^2}y}{\mathrm {d}x^2}\big|_{(x=-\frac{b}{2a})}=2a$ which is positive or negative ...
2
votes
1answer
114 views

Please help me solve this $x(\sqrt{2x+5}+\sqrt[3]{7x+13}) = 3x+6$

Wolfram Alpha shows that the answer is $x=2\,$ and $x=-2\,$ but what would be the best way of simplifying this equation ? It has been many years since I was in school , and I just cannot wrap my head ...
1
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2answers
39 views

prove of sum of n real numbers greater than n given product is one [duplicate]

If the product of $n$ positive real numbers is $1$.Then prove that their sum is never less than $n$.
0
votes
0answers
38 views

What is the formula called that calculates the scales of things?

I use this formula all the time: $$\frac{10}{100} = \frac{100}{x}$$ For example, the question may be, "if 10 apples equal 100 slices than 100 apples equals how many slices?" Is there a name for ...
-2
votes
1answer
27 views

Changing forms with factoring expressions [closed]

I need to know how to change between the quadratic forms: $ax^2+bx+c$ and $m(x+a)(x+b)$ by factoring trinomials given the values of $a, b$ and $c$ in the first form ($a$ and $b$ are not equivalent in ...
1
vote
2answers
62 views

How to solve for log with a number outside?

$$\log_6(4x-10)+1 = \log_6(15x+15)$$ This is a sample problem. I know that when the bases of log are the same, all you have to do is set the parenthesis inside equal to each other. If the $1$ wasn't ...
0
votes
1answer
45 views

Solving irrational inequality

Given inequality $(x - 2)\sqrt{x^2 + 1} > x^2 + 2$, find it's solution as intervals. And I have problem solving it. So at first, both $\sqrt{x^2 + 1} > 0$ and $x^2 + 2 > 0$. That means, that ...
2
votes
3answers
58 views

expansion of a generating function

I found this formula in a book $$\sqrt{1-4x}=1-2\sum_{n=1}^{\infty}\frac{1}{n} {{2n-2}\choose {n-1}} x^n$$ How can I prove that?
0
votes
2answers
32 views

How far is it from $A$ to $B$

If the boat moves downstream in the river, then going from country $A$ to country $B$ takes about $4$ hours, whereas if it moves upstream, then the same journey takes about $5$ hours. What is the ...
2
votes
6answers
603 views

Result of subtraction either positive or zero [closed]

I need to perform a subtraction where the result must be either positive or zero. I will use some pseudo code to represent what I mean (like it is not clear enough but still). ...
3
votes
4answers
387 views

Exponential Equations with Fractions

I have had some issues with the following two equations: $$ \frac{3^{n-2}}{9^{1-n}}=9$$ $$\frac{5^{3n-3}}{25^{n-3}}=125$$ If anyone could work them out step by step that would be awesome. I ...
-1
votes
1answer
27 views

Solving an algebraic expression by taking common factors

First of all, I'd like to apologize for this question because I'm sure the answer is simple, I'm just a bit rusty on basic algebra skills after being out of practice for so long. I can't seem to work ...
1
vote
1answer
39 views

Prove $\Sigma_{k=0}^{n-1}\lfloor x+\frac{k}{n}\rfloor=\lfloor nx\rfloor$ , n is a Natural Number

Prove the following identity: $$\lfloor x\rfloor +\lfloor x+\frac{1}{n}\rfloor +\lfloor x+\frac{2}{n}\rfloor +\lfloor x+\frac{3}{n}\rfloor+...+\lfloor x+\frac{n-1}{n}\rfloor =\lfloor nx\rfloor$$ ...
1
vote
2answers
33 views

A system of polynomial equations of degree $2$ in two variables

I need to find an explicit solution of this system of polynomial equations of degree $2$ in two variables $x,\,y$: $$\begin{cases} p_1x^2+q_1y^2+r_1xy+s_1x+t_1y+u_1=0\\ p_2x^2+q_2y^2+r_2xy+s_2x+t_2y+...
0
votes
1answer
49 views

How might I solve the following equation

$$x * 0.98^{\sqrt x / 321868} = 9.46 * 10^8$$ I mean is there any way to do this algebraically? There's a single variable, but I don't know of any way to manipulate it such that I can get x on one ...
0
votes
0answers
42 views

In which quadrant of the circle does the angle of $90^\circ$ lie?

By definition and with an authoritative reference, in which quadrant or quadrants does $90^\circ$ lie? (There are non-authoritative references which answer the question, and a related question which ...
2
votes
2answers
50 views

Compound interest: how to use the textbook formula?

To derive a general compound interest formula we can say: $$A_1=A_0 + rA_0=A_0(1 + r)$$ $$A_2=A_0 + rA_0 + r(A_0 + rA_0)=A_0 + 2rA_0 + r^2A_0=A_0(1 + r)^2$$ and so on. In general: $$A_t=A_0(1 + r)^t$$ ...
0
votes
0answers
20 views

What is the relationship between a,b,c, and d that makes the equation equal?

What is the relationship between $a,b,c$ and $d$ that makes $$ acx + ad + b = cax + cb + d $$ I don't understand what is being asked, I can't isolate any variable. My only attempt was to eliminate $...
4
votes
2answers
45 views

Find the value of constant $k$ that makes this function continuous

The function is $$h(x)=\begin{cases} \dfrac{4x^2+5x-6}{x+2} & \text{if $x\neq-2$}, \\[6pt] 3x+k & \text{if $x= -2$}. \\ \end{cases}$$ After factoring the fraction I am left with $4x-3= -11$ ...
1
vote
1answer
43 views

Task with using of A Linear Operator

Population of slithy toves living in severe and adverse conditions, subject to the following rules: a) On average, only half toves survive the first year of life, half of the remaining survives in ...
1
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2answers
42 views

If $f(x)=2x-4$, What would the following graph of $3f(x+2)$ look like?

To do this problem I substituted the $$(x+2)$$ as the x in the equation $$ 3f(x+2)=2(x+2)-4$$ $$3f(x+2)=2x+4-4$$ $$3f(x+2)=2x$$ $$3f(x+2)/3=2x/3$$ when i graphed the equation I did not get the ...
1
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0answers
37 views

Create a matrix of four-dimensional space rotate counterclockwise by the angle π / 3 around the plane

Create a matrix of four-dimensional space rotate counterclockwise by the angle $\frac{π}{3}$ around the plane \begin{cases} x − y + t = 0,\\[2ex] y + z + t = 0 \end{cases} on the basis of unit ...
1
vote
4answers
75 views

Subtracting $\frac{(x+3)}{(x^2-1)} - \frac{(x-2)}{(x^2+2x+1)}$

$\frac{(x+3)}{(x^2-1)} - \frac{(x-2)}{(x^2+2x+1)}$ To solve the problem I first dissembled the equation on the denominator $ \frac{(x+3)}{(x-1)*(x+1)} - \frac{(x-2)}{(x+1)^2}$ I multiplied the ...
1
vote
1answer
44 views

Interval for the solutions of $\{x+1\}<x^2-2x$ where $\{x\}$ is the fractional part of $x$.

Find the interval(s) which contain solutions of $$\{x+1\}<x^2-2x$$ where $\{x\}$ is the fractional part of $x$. I was told that one way of solving this would be graphically. However I generally ...
0
votes
1answer
61 views

Does $\lim_{n \rightarrow \infty}\dfrac{x^n}{n^x}$ exist? [closed]

I was solving a question on convergence and came up with $$\lim_{n \to\infty}\frac{3^n}{n^2}$$ My question is this: Does the following exist? $$\lim_{n \to \infty}\frac{x^n}{n^x}$$
1
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3answers
105 views

Solutions of $\lfloor 4x\rfloor+\lfloor 3x\rfloor=1$

Find all solutions of $$\lfloor 4x\rfloor+\lfloor 3x\rfloor=1$$ I have no idea as to how to go about this question. I would be grateful if somebody would please show me how to solve such questions. ...
0
votes
3answers
63 views

Simplify $\Bigg(\frac{x^3}{6} + \frac{1}{2x}\Bigg)\sqrt{1+\Bigg(\frac{x^2}{2} - \frac{1}{2x^2}\Bigg)^2}$

How is this integral simplified as shown? \begin{align} S &=2\pi \int^3_\frac{1}{2} \Bigg(\frac{x^3}{6} + \frac{1}{2x}\Bigg)\sqrt{1+\Bigg(\frac{x^2}{2} - \frac{1}{2x^2}\Bigg)^2} \mathop{\mathrm{d}...
1
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0answers
21 views

How to solve complicated floor function equations?

I have been thinking about solving equations involving several groups of floor functions. My research on solving floor function equations has only shown sums of individual floors, such as: $\lfloor ...
0
votes
1answer
31 views

Get a matrix of polynomial coefficients from the roots

I've got the polynomial $P(z) = \Phi_0 - \Phi_1z $ defined by the following matrices of coefficients: $$ \begin{eqnarray} \Phi_0 = \left[ \begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0.2 & 1 &...
1
vote
3answers
68 views

why cannot the limits be $-1$ and $-2$

I came across a problem in definite integral as : Evaluate $$I=\int_{0}^{3} x\sqrt{1+x}\:dx$$ By the substitution $1+x=t^2$ so book has given lower and upper limits as $t=1$ and $t=2$ which is ...
1
vote
4answers
44 views

how many answers do this equation have$3^{2x}-34(15^{x-1})+5^{2x}=0$

How many answers do this equation have? $3^{2x}-34(15^{x-1})+5^{2x}=0$ My Attempt:$3^{2x}+5^{2x}=34(15^{x-1})$.Now what to do?
0
votes
2answers
36 views

Algebraic simplifications

Given the following line of maths: $$(1-x)(1+x)(1+x^2)(1+x^3)...(1+x^{2^n})$$ we can simplify it: $$(1-x^2)(1+x^2)...(1+x^{2^n})$$ and so on to obtain: $$(1-x)(1+x)(1+x^2)(1+x^3)...(1+x^{2^n}) = (...
5
votes
2answers
74 views

Condition on $a$ for $(x^2+x)^2+a(x^2+x)+4=0$

Find the set of values of $a$ if $$(x^2+x)^2+a(x^2+x)+4=0$$ has $(i)$ All four real and distinct roots $(ii)$ Four roots in which only two roots are real and distinct. $(iii)$ All four imaginary ...
0
votes
1answer
20 views

How to convert a function to the form y = A sin(Bx + C) + D to find the phase shift, period, and frequencies?

The question asks us to find the period of the function $y=\sin(√2x) + \sin(3√2x)$. I usually know how to find the period and all that in the format of $y = A \sin(Bx + C) + D$, but how do I get this ...
1
vote
1answer
15 views

find a real number in a form

I solve this problem by assuming the real number is $x$, then I got $m^2+xm-(2x+4)=0$. Then I solve the equation for $m$. This is complicated. Is an easy to find the real number? What real number ...
1
vote
1answer
41 views

$(a_{1},a_{2})$ and $(b_{1},b_{2})\in \mathbb{Z}^{2}$ with certain property

Can we find $(a_{1},a_{2})$ and $(b_{1},b_{2})\in \mathbb{Z}^{2}$ such that $$a_{1}a_{2} - 6\cdot b_{1}b_{2} = 2$$ and $$a_{1}b_{2}+a_{2}b_{1} = 1$$ $\textbf{Note.}$ $a_{1},a_{2},b_{1},b_{2} \neq 0$.
4
votes
2answers
72 views

If $x$ and $y$ are positive numbers less than $20$ for which $x+y+xy=76$, what is $x+y$?

What is a simple way to solve this problem? I can do it by trying $x$ and $y$, starting from $1$. That does not look like the best way. If $x$ and $y$ are positive numbers less than $20$ for which ...
0
votes
0answers
21 views

sufficient conditions for: homogeneous polynomial non-negative for $x\geq0$

Are there any sufficient conditions, s.t. a homogeneous polynomial with real coefficients (of degree $m$ in $n$ variables) is non-negative for all $x\in\mathbb R^n$ with $x\geq0$? $x\geq0$ here means ...
0
votes
1answer
28 views

Solutions of quadratic equation with n variables

I'm trying to find the roots of a quadratic equation with $n$ variables. I've looked through the internet but I wasn't able to find any convincing formula. Given a vector $v=${$x_1, x_2, x_3, ..., ...
1
vote
1answer
64 views

Can we logically analyze mathematical theorems as if-then statements?

Many theorems in math have an if-then form. For example: "If a polynomial is of $n^{th}$ degree, then it has $n$ roots. In my other question, I learned that in order to analyze statements using truth ...
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votes
1answer
22 views

Determining values of the statements

I have a hard time, solving some logical statement exercises. Given two statements $v(x): |x| = 2$ and $u(x): x > 1$, where $x \in A = \{-1, 0, 1, 2, 3, 4\}$, I have to determine all values of ...
0
votes
0answers
26 views

How to determine value of statement

Given statement $v(x): |x| = 2 \text{ where } x \in A = \{-1, 0, 1, 2, 3, 4\}$. What will value of $\lnot v(x)$ look like?
0
votes
2answers
69 views

Japanese Test Sample Question

Sorry for the ambiguity in the title, I had no idea what else to put. I am studying for a test I have to take for a Japanese scholarship and I have been able to do all of the other sample questions ...
5
votes
1answer
34 views

Value of $k$ for equation to have no solution

What are different integer values of $k$ between $1-9$ for which the equation $$|x-1|+|x-2|+|x+1|+|x+2|=4k$$,has no solutions. Now there are 24 different ways of having signs ie the equation after ...
0
votes
1answer
20 views

If roots of a polynomial are complex number how to visualize Geometrically

I have one basic doubt in complex numbers. We know that if a polynomial equation $P(x)=0$ cuts the $X$ axis or touches the $X$ axis, then they represent Real roots of the polynomial or real roots with ...
0
votes
0answers
13 views

Algebra with whole no. 1

$1 - ( x + 1 / 2 ) + ( X - 2 / 6) $ My working $ 1 - ( 6x + 6 + 2x - 4 / 12 ) = 12/12 - 8x+2 / 12 = -8x+10 / 12 = -4x + 5 / 6 $ Why is my working wrong ? Thanks in advance !
-3
votes
4answers
43 views

Algebra Problem regarding powers [closed]

if $a \gt b$ prove that $a^3+2ab^2\gt b^3+2a^2b$
0
votes
1answer
63 views

Can someone explain how this

$$ x\left\lbrace 4\left[ 1+\left(\frac{x}{2}\right)^{2}\right]\right\rbrace^{-1/2} = \frac{x}{2}\left(1 + \frac{x^2}{2^2}\right)^{-1/2} $$ This fact was shown in my textbook with no work and I can't ...
1
vote
1answer
20 views

graphing multiple functions limited between between two x values

On a graph you can limit the showing of a function between two x values by multiplying by a complex when outside. $$\\f(x) = ax * \frac{(\sqrt{(max -x)} * \sqrt{(x - min)}}{(\sqrt{|(max -x)|} * \sqrt{...
0
votes
3answers
37 views

Why is solution to this inequality equal to $\mathbb{R}$?

Given inequality $$-5(1- x)^2 < 3x + 11$$ which after algebraic manipulation looks like $$5x^2 - 7x + 16 > 0,$$ it is obvious that it's discriminant is equal to $\Delta = -271$. In my book it is ...