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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1answer
51 views

How to rationalize this root form?

Suppose that we have a equation like this: $$\sqrt{a+b+2\sqrt{ab}}$$ or $$\sqrt{a+b-2\sqrt{ab}}$$ In order to rationalize it, we can apply the formula: $$\sqrt{a} + \sqrt{b} = \sqrt{a+b+2\sqrt{...
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2answers
41 views

System of Equation Problem? 52x + 16y = 100

I have forgotten how to solve this type of problem. $$52x + 16y = 100$$ Is this the method? Substitute $0$ for $x$ $$0+ 16y = 100\\y= 100/16 \\y= 6.25$$ Substitute $0$ for $y$ $$52x+0 = 100\...
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2answers
221 views

Chapter 3, question 20 part b of Spivak's Calculus 3rd edition

Suppose that $f(y)-f(x)\le(y-x)^2$ for all $x$ and $y$. (Why does this imply that $\lvert f(y)-f(x)\rvert \le (y-x)^2$ ?) .Prove that $f$ is a constant function. Hint: Divide the interval from $x$ to $...
3
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1answer
60 views

Special representation of polynomial

How to prove that for natural $n$ the polynomial $(x^4-6x^2+1)^n$ can't be represented in such a way $$ (x^4-6x^2+1)^n=f(x)^2+1, (x^4-6x^2+1)^n=g(x)^3-1, $$ where$f(x), g(x)$ are polynomials. ...
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1answer
85 views

Why does the borrowing method for subtracting stop working if the bottom number is bigger?

My brother was given the problem $2.3-4$, and tried to solve it using the standard one over the other format. $.3-.0=.3, 2-4=-2$, answer is $-2.3$. He looks at the answer in the back and sees that it ...
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3answers
55 views

One over 2 times itself?

The question asks to simplify the following: $\pi + 2/(\pi+ 2/(\pi + 2/( ...$ repeats Having difficulty seeing the reduction of this. Any ideas? The answer to the question seems to indicate ...
2
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1answer
92 views

$x^4-2x^3+x=y^4+3y^2+y$ in the set of integers

The task is to solve the equation $x^4-2x^3+x=y^4+3y^2+y$ in integers. I expect is has something to do with factorizing but have no concrete idea; any help? thx guys
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4answers
124 views

Solve this algebra problem in terms of $x$ and $y$

Solve the equation for $x$ and $y$ give $xy+8x+y=83$. I did mine by isolating factoring $x$ and got $$x = \frac{83-y}{y+8}$$ Is it correct?
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1answer
177 views

Find the all possible real solutions of $x^y=y^x$ [duplicate]

Find the all possible real solutions of $$x^y=y^x$$ $x,y$ both are real numbers. My attempt:I observed the following solutions $x=2,y=4$ $x=4,y=2$ $y=x$ Is there any other possible solutions?
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2answers
2k views

When does the equality hold in the triangle inequality? [duplicate]

Hi guys could you please help me on this question I'm confused. question: when does the equality hold in the triangle inequality: my attempt : $|x + y| \leq |x| + |y|$ this implies $(|x+y|)^2 = (|...
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2answers
39 views

Show that two consistent systems are equivalent to each other [duplicate]

$A: n \times n$, $B: n \times m$ and $A$ is invertible. Show that "$\forall \vec{b} \in \mathbb{R}^n, B \vec{x} = \vec{b}$ is consistent" is equivalent to "$\forall \vec{b} \in \mathbb{R}^n, (...
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2answers
287 views

Solved a quartic equation by dividing

I was asked to solve: $$x^4+2x^3-22x^2+2x+1 = 0$$ Without using differential calculus (Newton's Method). My Progress: Dividing by $x^2$, I get: $$x^2+2x-22+\frac{2}{x}+\frac{1}{x^2} = 0$$ $$x^2 +\frac{...
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1answer
194 views

Parabola - How far from the thrower does the ball strike the ground?

The height of a ball thrown in the air is given by $h(x) = \frac {–1}{12} x^2 + 6x+ 3$, where x is the horizontal distance in feet from the point at which the ball is thrown. c. How far from the ...
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2answers
157 views

For which values of $b,c$ is the matrix invertible?

$A =\left(\begin{array}{ccc} 0 & 1 & b \\ -1 & 0 & c \\ -b & -c & 0 \end{array}\right)$ I tried to come up with the RREF($A$), but in the final step I have the matrix: $A =\...
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4answers
80 views

There exists a real number $x$ such that if $x^2 ≥0$ then $ x=0$.

I have to prove: There exists a real number $x$ such that if $x^2 ≥0$ then $x=0$. I have no idea what should I proceed. I tried to come up with the contrapositive, and it doesn't help. I have this ...
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1answer
276 views

Find a basis for a subspace (working included)

I have been working on this question and I am not too sure if it is correct or not. Any help would be appreciated. Question (in picture format): http://i.imgur.com/E4MhH99.png My working: The first ...
3
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1answer
127 views

Solving equations: reasoning doesn't work backwards?

In doing my (high school) math homework, I came to an issue that doesn't make sense to me. Given an equation $0 = a_1 + a_2x + a_3x^2 + \dots$, we can multiply both sides by $x$ to obtain $0 = a_1x + ...
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2answers
73 views

How do you show $(ac)^2+(bd)^2+(bc)^2+(ad)^2=1$

Show $(ac)^2+(bd)^2+(bc)^2+(ad)^2=1$ if $a^2+b^2=1$ and $c^2+d^2=1$. I was thinking about solving for variables and plugging in, but that seems like too much work. Is there a simple trick or ...
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2answers
26 views

System of equations, how to do this

This is for a physics class, but I think the question is mathematical in nature. We have the following equations: $$ \mu (c-c') = -m(v-v')$$ $$ \mu (c^2 - c'^2) = -m(v^2-v'^2) $$ $$ c+ c' = v + v'$$...
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2answers
98 views

Solving an equation with rational function under square root

I am trying to solve for D in the equation below and keep coming up with different answers. I can hopefully take it from there. $$A=B\sqrt{1-\frac{2C(\frac{D}{E^2})}{FG^2}}$$ Thank you
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2answers
99 views

How to prove this equation? $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = \frac{1}{a+b+c}$

Suppose $a, b$, and $c$ are nonzero real numbers which satisfy the equation: $\frac{1}{a}+\frac{1}{b}+\frac{1}{c} = \frac{1}{a+b+c}$ Prove: if $n$ is an odd integer, then $a^n + b^n + c^n=(a+b+c)^n$ ...
0
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1answer
31 views

Find the maximum value of a for which there is a real number solution

Equation 1: $x^2+y^2=1$ Equation 2: $x^2•y^2=a$ I am sure the normal way of finding a solution won't work for this question. Because when I simplified , I got $ a=x^2-x^4$. Can you solve without ...
3
votes
3answers
101 views

Difficulty in “expressing radical as square”

I have to get from this expression: $(4+2\sqrt3)(\sqrt{2-\sqrt3})$ To this expression: $\sqrt2+\sqrt6$ I tried to square $(4+2\sqrt3)$ and put it inside the radical, so: $\sqrt{(16+12+16\sqrt3)(2-\...
1
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1answer
82 views

Decompose to even and odd functions

Suppose we have the function $f(x) = |x-1|$. I have to find the even and odd parts of the function and write them in terms of Heaviside Function. I have no idea what should I do here? I tried and it ...
3
votes
1answer
105 views

Is this AM-GM application correct?

I have got an inequality down to proving that: if $a,b,c$ are positive reals that satisfy $a+b+c=1$, then $$\frac{1}{1-\sqrt{a}}+\frac{1}{1-\sqrt{b}}+\frac{1}{1-\sqrt{c}}\ge \frac{9+3\sqrt{3}}{2}$$ ...
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3answers
203 views

Problem 17 in chapter 3 of Spivak book

If $f(x)=0$ for all $x$, then f satisfies $f(x+y)=f(x)+f(y)$ for all $x$ and $y$, and also $f(xy)=f(x)f(y)$ for all $x$ and $y$. Now suppose that $f$ satisfies these two properties, but that $f(x)$ is ...
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1answer
41 views

How did the author change the inequality into an equality?

While trying to prove that the function $f(n)=8x+128$ is $\mathcal{O}(n^2)$, the author turns an inequality into an equality. This is what he did: We wish to show that $f(x) = \mathcal{O}(n^2)$. ...
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2answers
345 views

I can do the math but not the problems, help? [closed]

So whenever my instructor / teacher is going over the notes and teaching the new lesson to the class, I listen to what he says, take notes, and do the practice problems along with him. Often times I ...
0
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2answers
82 views

Solution to simple algebra problem

I don't seem to be able to solve this for $x$: $$y = \frac{e^x + e^{-x}}{2}.$$ Ans. is $$x = \ln\left(y\pm\sqrt{y^2-1}\right),$$ but I'd appreciate seeing the intermediate steps. Thanks.
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2answers
212 views

Find the locus of points whose distances from the line $y=\sqrt3x$ and x-axis are equal.

Find the locus of points whose distances from the line$\hspace{0.2cm}$ $y=\sqrt3x$$\hspace{0.2cm}$ and x-axis are equal. My solution:I start with the following $$\frac{|\sqrt3x_1-y_1|}{2}=\frac{|y_1|}...
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2answers
59 views

Seeking proof using mathematical induction

\begin{equation}a: \mathbb N ×\mathbb N \to \mathbb R \end{equation} where for all \begin{equation}x,y\in\mathbb N\end{equation}\begin{equation}a(x,y) =a(y,x)\end{equation} How do I show that the ...
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1answer
31 views

Find the intersection between a linear equation and a quadratic equation, where the linear equation is x, not y

What the title says essentially. Where one is x=0.6(y-4)^2+4 and another is x=15 The confusing part for me is that it is x=15, there is no y. There are a lot of help for y=?, but x=? NEVERMIND, I ...
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3answers
1k views

Proving a function has real roots

I am not interested in finding roots but interested in proving that the function has real roots. Suppose a function $f(x) = x^2 - 1$ This function obviously has real roots. $x = {-1, 1}$ How could ...
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1answer
59 views

Is it possible to find integer solutions for this equation?

I just thought of something really crazy off the top of my head, $$(2+3^{m/n})^{n/m}=(3+2^{n/m})^{m/n}.$$
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1answer
43 views

simplifying expression of roots of cubic equation

I came across with this question about roots of polynomials. Suppose $a$, $b$ and $c$ are the roots of $x^3-4x+1=0$. Find the value of $ \frac{a^2bc}{a^3+1}+\frac{ab^2c}{b^3+1}+\frac{abc^2}{c^3+1}. $ ...
0
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1answer
28 views

Working with sets and its laws

Given: $(A \oplus B) \cup C = (A \cap C) \oplus ( B - C )$ Work with algebra of sets to prove the proposition above is true. In order to give a solution to this problem I've done the process ...
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2answers
313 views

The width of a rectangle is 3 less than twice its length. If the area of the rectangle is 131 cm^2, what is the length of the diagonal?

The width of a rectangle is 3 less than twice its length. If the area of the rectangle is 131 cm^2, what is the length of the diagonal? I set up the basic equation to solve for l and I know I need ...
2
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2answers
47 views

Can the following equations be solved without the need of numerical methods?

I'm taking advanced algebra in school. I have been asked to solve two equations: $\log_{6}(1-x) + \log(x^{2}-9) = 2 \\$ $ 3^{x+2} + 2^x = 5 $ The teacher said this equations can be solved ...
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3answers
297 views

Solve the equation

$$9^x=5^x+2\sqrt{20^x}+4^x$$ I'm not really sure where to start. I tried simplifying with logarithms and factoring out the x but it ended up looking just as complicated..
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4answers
135 views

Not understanding how to factor a polynomial completely

$$P(x)=16x^4-81$$ I know that this factors out as: $$P(x)=16(x-\frac { 3 }{ 2 } )^4$$ What I don't understand is the four different zeros of the polynomial...I see one zero which is $\frac { 3 }{ 2 ...
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3answers
185 views

Under what conditions will one solution of $ax^2+bx+c = 0$ be the reciprocal of the other?

Under what conditions will one solution of $ax^2+bx+c = 0$ be the reciprocal of the other?
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2answers
37 views

Stuck finding the zeros of a polynomial (complex and real)

Stuck finding the zeros of this polynomial (complex and real): $$x^4+2x^2+1$$ I am not sure how I would factor this. The constant value is really throwing me off. I just need a hint on how to get ...
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1answer
25 views

Transformations of function not vertical/horizontal

I do not understand how to solve these transformation of functions. Can someone please explain how these are solved? http://i.imgur.com/uWLhxJP.png and http://i.imgur.com/HeytcX2.png Thank you
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2answers
49 views

$64 = a^2-b^2$, how many solutions are there

$64$ was written as a sum of two squares of two natural numbers. How many solutions does this equation have? $$64 = x^2-y^2 = (x+y)(x-y)$$ For example this works for $10$ and $6$, but how can I ...
0
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1answer
63 views

System of equations with parameter

I have been trying to solve this problem for a week now. It goes like this: Find all values of $a$ for which the system $$ \begin{cases} x^2-2x+y^2 = 1 \\[1ex] \dfrac{x+|x|}{y-a}=2 \end{cases} $$ has ...
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2answers
69 views

Complex power of a real number

What is the meaning of $(-1)^{i}$, where $i^{2}=-1$ and what is its value?
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3answers
131 views

How do I show algebraically that the period of the tangent function is $\pi$?

How do I show that the positive real number $p$ for which $\tan (x+p)=\tan (x)$ is equal to $\pi$? In essence how do I prove the period of the tangent function is $\pi$? Please bear in mind I am a ...
1
vote
1answer
34 views

Integer solutions for equation with two variables

Can someone explain how wolfram alpha calculates integer solutions for these kind of equations: $$ m=\frac{681+13973k}{2021} $$ and how can I do this myself on the paper? Here is a link to the ...
0
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1answer
40 views

Polynomial function theorem

The following is from Spivak chapter 3 page 49: Prove that for any polynomial function $f$, and any number $a$, there is a polynomial function $g$, and a number $b$, such that $f(x)=(x-a)g(x)+b$ for ...
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3answers
85 views

Rearranging the polynomial $x^3-23x^2+142x-120$ prior to factoring it

In the example 15: They are saying that, $$x^3-23x^2+142x-120 = x^3-x^2-22x^2+22x+120x-120$$ From where did $22x^2$ and $22x$ come and also $120x$. Please help me clear my confusion.