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
79 views

How is the discriminant is able to find coefficients in a quadratic equation?

I know how to solve for $k$ in $kx^2-30x+25=0$ using $b^2-4ac$, but i want to know how the discriminant does this. How are we able to just plug the coefficients into the discriminant and get the ...
0
votes
2answers
24 views

Solving for Variable in Rational Expressions

Hi I have a really simple issue, I am doing a circuit problem and I get a final equation: $$\frac{v_1-12}{2\ k\Omega}+\frac{v_1}{k\Omega}+\frac{v_1}{r_1}=0$$ Im trying to get $v_1$ by itself, the ...
3
votes
1answer
125 views

Prove that $\pi > 24\small{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}}$

Prove that $\pi > 24\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$. I tried using trig but I couldn't solve it. A hint I was given is to use half angle identities. This should be easy for someone who is ...
0
votes
1answer
45 views

Solving functional equation $f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n).$

Given natural number $n, k$ consider nondecreasing function $f:\mathbb{N}\cup {0} \to \mathbb{N}\cup {0}$ such that $$ f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n), $$ for ...
1
vote
1answer
30 views

Complete the square to change into standard form

Here is the equation: $x^2 + y^2 + 4x - 6y - 3 = 0$ Here are the instructions: Complete the square to change the equation info standard form. Then graph the equation. Because both $y^2$ and $x^...
2
votes
2answers
481 views

Why is slope rise/run?

What makes slope rise over run? What makes the standard equation for a line use a slope of rise over run as opposed to run over rise? What would the standard equation of a line look like if m was ...
7
votes
2answers
770 views

Reverse the equation to solve for y

I'm trying to solve for $Y$ in this equation: $\frac {(X-Y)}{Y} = Z.$ I tried applying some of the answers from other questions but I'm having special trouble with figuring out how to get the $Y$ ...
3
votes
1answer
92 views

Infinite product: $(1-0.5^2)(1-0.5^3)(1-0.5^4)…$

Find a closed form for the value of the infinite product $(1-0.5^2)(1-0.5^3)(1-0.5^4)...$ I know it converges. At first I thought it was the Euler–Mascheroni constant, but it's only accurate to about ...
0
votes
2answers
56 views

ordered triplets of integer $(x,y,z)$ in $z!=x!+y!$

$(1)\;:$ How many ordered triplets of positive integers $(x,y,z)$ are there are such that $z! = x!+y!$ $(2)\;:$ How many ordered triplets of positive integers $(x,y,z)$ are there are such that $w! =...
0
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2answers
60 views

Given a function $f(x)$, determine if the following function is even, odd, or neither

Given a function $f(x)$, determine if the following function is even, odd, or neither: $g(x) = −2[f(x)+f(−x)]$ I understand that a function is even if $f(-x) = f(x)$ is even, and odd if $f(-x) = -f(x)...
0
votes
1answer
14 views

Graph $\ell(x) = 3x +2$. What would $\ell(x+1) - \ell(x)$ be for all $x$?

What does it mean to give an answer "for all $x$"? Is it asking for the domain?
1
vote
2answers
427 views

How to prove that a quadratic equation implies both variables are zero

This might be really simple, but I can't find how to prove that $a^2-\frac{2}{3} ab+b^2=0$ implies that both $a$ and $b$ are zero. Any help will be appreciated!
1
vote
1answer
165 views

Solving System of Equations using transformation rotation

I've never had to post the same question twice, but my last post is getting filled out with work and I'm going about it a different way so I figured i'd try a whole different question So This is the ...
1
vote
1answer
40 views

Finding the alternate forms of ratios

This is a super basic algebra question but I can't figure out how you get the alternate form of: $$\frac ab = \frac cd$$ Which is: $$\frac {a+b}b = \frac{c+d}d$$ The process explaining how we ...
0
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2answers
53 views

Recursive Definition notation example

In my textbook, the author shows this example of recursion, and I can't make heads or tails of it. Can someone give a better explanation of this...
1
vote
2answers
66 views

Understanding $3^n < n!$

In my class, we are given the answer to this proof. I understand how the inequality was simplified, but don't understand why the bolded statement is true for $k+1,$ or more simply, how that proves by ...
1
vote
1answer
36 views

Prove that $\bar z_1 z_2+z_1 \bar z_2=2\Re(\bar z_1 z_2)$

Let us consider $z_1, z_2\in \mathbb C$; we have: $$\bar z_1 z_2+z_1 \bar z_2=2\Re(\bar z_1 z_2)$$ it is easy to prove if we put $z_1=x_1+iy_1$ and $z_2=x_2+iy_2$. But suppose we do not want to use $...
1
vote
2answers
130 views

Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
1
vote
1answer
128 views

Looking for resources for understanding derivation of demand from utility

I am struggling with my homework and would very much appreciate a rundown of the math or pointers to where I can find help otherwise. Quoting from the assigment: There are $n$ sectors in the ...
0
votes
2answers
46 views

$2k^2+7k=2mk+3m+36$. Find all non-negative integer solutions.

I've tried this: $2k(k-m)+7k-3m-36=0$. And I'm stuck. How do I solve this one?
17
votes
2answers
444 views

Trig sum: $\tan ^21^\circ+\tan ^22^\circ+…+\tan^2 89^\circ = ?$

As the title suggests, I'm trying to find the sum $$\tan^21^\circ+\tan^2 2^\circ+...+\tan^2 89^\circ$$ I'm looking for a solution that doesn't involve complex numbers, or any other advanced branch in ...
3
votes
2answers
7k views

Exponential equations with variables on both sides

I have the following: $$8^{3x+4} = 5^{4x-2}$$ How would I solve this? I tried this: $$(3x+4)\log 8 = (4x-2)\log 5$$ but have no idea where to go from there. Thank you!
3
votes
2answers
154 views

Trigonometric equation $\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X)$

A trigonometric equation is to be solved, the solution ($X=10^\circ$) is very clear but I need a proper method $$\sin(60^\circ-2X)\sin(5X)=\sin(8X)\sin(3X).$$
0
votes
1answer
61 views

Multiplying a factorial with non factorial

I'm trying to understand the following equation, do I multiply the 2 and the 1 to get (n+2)! ? $$(n+2)(n+1)! = (n+2)!$$
1
vote
2answers
72 views

$x^2-xy-2x+3y=11$. Find natural solutions.

I've got this factorizing: $(x-2)(x-y)=11-y$. And I'm stuck on it. How can I solve it?
0
votes
1answer
55 views

Adding and Subtracting numbers with exponents

Why is $2^{k+1} + 2^{k+1} = 2^{k+2}$ and not $4^{k+1}$
0
votes
1answer
25 views

$2x^2+2xy-x+y=112$. Find natural solutions.

I've got this and I'm stuck: $(x-2)(x-y)=11-y$. Is it possible to make something out of this? Oops, I've got that on the other problem.
1
vote
4answers
83 views

If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots $ n terms… [closed]

Question( from sequences) : If $1(0!)+3.(1!)+7(2!)+13(3!) +21(4!) + \cdots $ n terms = $(4000)(4000!)$ Then what is the value of n. How to proceed in this please suggest , will be of great help to ...
1
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2answers
44 views

What is the sum $\sum_{r=1}^\infty \frac{r}{4r^4+1}$ equal to?

Problem : If $$T_r =\frac{r}{4r^4+1}$$ then the value of $$\sum^{\infty}_{r=1} T_r$$ is ? How to start such problem I am not getting any clue on this please suggest thanks .
1
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2answers
59 views

The equation $5^x+2=17^y$ doesn't have solutions in $\mathbb{N}$

Problem: Prove that the equation $5^x+2=17^y$ doesn't have any solutions with $x,y$ in $\mathbb{N}$. I've been analyzing the remainder while dividing by $4$, but I'm getting nowhere.
1
vote
2answers
50 views

Evaluate this square root

$\sqrt{6 + 2\sqrt{5}} + \sqrt{6 - 2\sqrt{5}}$ I have no clue where to begin. I would appreciate a hint, the answer should be $2\sqrt{5}$ In general, how do you evaluate $\sqrt{a + b} + \sqrt{a - ...
-1
votes
6answers
122 views

Which of the following is equivalent to the expression? $i^{22}$

Which of the following is equivalent to the expression? $i^{22}$ A.) $-1$ B.) $i$ C.) $1$ D.) $-i$ What is $i$? How could it have a exponent if it's an imaginary number?
1
vote
6answers
51 views

Solve a linear function

How do I solve this homework assignment? For a linear function $y=f(x)$, $f(-3) = 25$ and $f(3) = 11$. Determine $f(-20)$. I know that with the values $f(-3) = 25$ and $f(3) = 11$ I am suppose to ...
2
votes
1answer
69 views

If $ab+bc+ca=1$, then $\frac{((a+b)^2+1)}{(c^2+2)}+\frac{((b+c)^2+1)}{(a^2+2)}+\frac{((c+a)^2+1)}{(b^2+2)} \geq 3$

Let $\displaystyle a, b, c> 0, ab+bc+ca=1$. Prove that the following inequality holds: $$\frac{((a+b)^2+1)}{(c^2+2)}+\frac{((b+c)^2+1)}{(a^2+2)}+\frac{((c+a)^2+1)}{(b^2+2)} \geq 3.$$ I tried ...
3
votes
2answers
119 views

Integral difficulties (attempt included)

I am having difficulties with the following integral. I began working on it and thought I had obtained the answer, but when I went to graph it I received an integral of 1. I obtained the same answer ...
1
vote
1answer
76 views

Functional equation- solving techniques

I'm basically a total novice with functional equations and have some questions regarding the solving technuiqes of them. Although, i'm adware of the lack of general solving methods, I have noticed ...
2
votes
2answers
109 views

Triangle inequality problem with equality

How does one prove that, for any reals $x,y$ , there holds the equality $$|x|+|y|+||x|-|y|| = |x-y|+|x+y|\quad?$$ I have tried this using both the reverse and triangle inequalities, but I cannot get ...
3
votes
1answer
84 views

How do I solve $x=\log^e{(x+1)}$ analytically?

How do I solve the following, analytically? $$x=\log^e{(x+1)}$$ It looks like it should be simple, but whether I take the $e$th root of each side or take the $\log$ of each side (ending up with a $\...
-3
votes
2answers
47 views

How would I be able to tell if some vector is in the span of a set of vectors?

Given the following, how would I be able to tell if b and c are in the span of the set of vectors S? Any help is appreciated. enter link description here
1
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1answer
37 views

A simple problem of the equation of a plane.

Two planes given $$x-y+z=5 , \hspace{0.5cm}x+y+z=3 $$ Their intersection is a line $l$.Find the equation of a plane such that the line $l$ is perpendicular to that required plane and this plane ...
0
votes
0answers
123 views

How to show that if average of squares equals square of average then all $X_i$ are equal?

Defining $A(X) = \sum_{i=1}^np_iX_i$, how would I show that given $A(X^2) = A(X)^2$, all $X_i$ must be equal? I tried by contrapositive - assume they are not equal and show that $A(X^2) \ne A(X)^2$. ...
4
votes
5answers
126 views

Show that $\gcd(a,b)>1$

Given are three natural numbers $a$, $b$ and $c$, for which $$\frac1a+\frac1b=\frac1c,$$ show that $\gcd(a,b)>1$. Could you someone provide a hint? I already tried algebraic manipulation, but ...
1
vote
2answers
58 views

integer $n$ for which $n^6+3n^5-5n^4-15n^3+4n^2+12n+3$ is a perfect Square

Prove that the no integer $n\;,$ for which $n^6+3n^5-5n^4-15n^3+4n^2+12n+3$ is a perfect Square. My Try:: We can write $(n^6+3n^5-5n^4-15n^3+4n^2+12n+3) = (n^3+an^2+bn+c)^2$ Now Here we have to ...
0
votes
1answer
20 views

How do I solve for x and y: x + 0.0467 + y = 1.000?

I am trying to find the isotopes for percent abundance question. I am looking over and answer and can't figure it out because of the math. Here it goes. 1) Set up a system of two equations in two ...
0
votes
1answer
54 views

find minimum and justify it

After having found the derivative of which if i am not mistaken is : I need to find the minimum of the function for which if I am not mistaken I equal to 0 the first derivative is this the right ...
2
votes
1answer
68 views

Partial fraction help

I need Help figuring out how to solve the indefinite integral of $$\int{ -5x^3-2x^2+32\over x^4-4x^3 } dx $$ using partial fractions. Please help. Thank you! I have already checked the online ...
0
votes
1answer
66 views

Value of $x$ where the graph lies below the graph of f(X)

From this question (How can I find the values of $x$ where a function lies below or above the axis?) I learn that "lies below" means $f(x)<0$ , now my question is , how can I check values that lie ...
0
votes
1answer
64 views

How can I find the values of $x$ where a function lies below or above the axis?

Let's imagine this problem: Find the values of $x$ where the graph of $$f(x)= \frac{3x^2}{x^2-1}$$ lies below the $x$-axis. I know how to find the intercept $(0,0)$, but I don't understand what ...
0
votes
3answers
39 views

no. of mapping from from $A\rightarrow B$ such that $f(i)<f(j)\;\forall \; i<j, $ is

If $A = \left\{1,2,3,4\right\}$ and $B = \left\{1,2,3,4,5\right\},$ Then $(a)\; :: $ Total no. of mapping from from $A\rightarrow B$ such that $f(i)<f(j)\;\forall \; i<j, $ is $(b)\;\;::$ ...
0
votes
1answer
71 views

Equation $ \sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{a_k}=\sqrt[n]{b_1}+\sqrt[n]{b_2}+\cdots+\sqrt[n]{b_l} $

Let $k,l$ be natural numbers and $\{ a_i, b_i \}$ be real positive numbers such that $a_1\leq a_2 \ldots \leq a_k$, $b_1\leq b_2 \ldots \leq b_l$ and $$ \sqrt[n]{a_1}+\sqrt[n]{a_2}+\cdots+\sqrt[n]{...