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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1
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0answers
23 views

What is the difference between the slope and the angular coefficient?

What is the difference between the slope and the angular coefficient?
1
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1answer
22 views

Area of “Scalene” rectangles

My friend told me that from antiquity land revenue officials compute area of "reasonably" rectangular ( diagonal no matter) fields to assess tax taking the opposite sides average as .. $$ A = ...
0
votes
1answer
20 views

Hearts Game Word Problem

Hearts is a game where the lowest score wins. We know this : The fourth player scored a $105$ The first three players scored a combined value of $103$ No scores are zero No score (except loser) can ...
0
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2answers
28 views

Find the height of the box.

A box contains 150 candles. It has a width that is five times the height of the box and a length that is thrice the width of the box. The volume of each candle is 1 $in^3$. Find the height of the box. ...
0
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3answers
40 views

Give a serious explanation of the difference between an equation and a function.

What's the difference between an equation and a function? I mean, I am not seeking for a high school-like answer like "an equation has an equals sign". I want to know what is the fundamental ...
-1
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1answer
30 views

How many different three-digit house numbers could be made?

a shopkeeper sells house numbers. she has a large supply of the numerals 4, 7 and 8, but no other numerals. how many different three-digit house numbers could be made using only the numerals in her ...
6
votes
3answers
426 views

Is there an algorithm to compute the degree of a polynomial?

Let $f\in k[X]$ be a polynomial in one unknown over any field (or any nice enough commutative ring, I imagine - it shouldn't matter) and suppose that all we can do to understand $f$ is to evaluate it ...
0
votes
2answers
20 views

How to calculate test score needed to maintain a certain average

I know the answer is probably very basic math, but I can't seem to figure it out. I want a 92 overall grade in math. -Test scores (make up 60% of grade): 86, 91, 90, 89 -Quiz scores (make up 25% of ...
-2
votes
1answer
35 views

The pascal triangle [on hold]

I really dont know what rules to apply to get this answer... but i know the following. I know that $$(a+b)^5$$ a decrease from $5$ to $0$ while $b$ increases .. eg \begin{array}{} ...
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5answers
42 views

Express $x$ algebraically with no nested radicals [on hold]

Given that $x= \sqrt{7+4\sqrt{3}}$, express $x$ algebraically with no nested radicals.
1
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2answers
63 views

How do I solve $2^x + x = n$ equation for $x$?

I need to solve the equation $$2^x + x = n$$ for $x$ through a programming-based method. Is this possible? If not, then what would be the most efficient way to approximate it?
0
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4answers
29 views

Points $A$, $B$, and $C$ are on the circumference of a circle with radius 2

Points $A$, $B$, and $C$ are on the circumference of a circle with radius $2$ such that $\angle BAC = 45^\circ$ and $\angle ACB = 60^\circ$. Find the area of $\triangle ABC$. I've drawn a circle ...
2
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2answers
35 views

Expressing a function in terms of compositions of three functions.

Express the function F in the form $f \circ g \circ h$. $$F(x)=\frac {9}{( x^2 + 7)}$$ I'm not sure how to get $x^2+7$ in the denominator. Here is what I tried: $$h(x) = (x+7)$$ $$g(x) = x$$ ...
0
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1answer
21 views

Difference quotient of $f(x)= 2-6x+4x^2$

I need to find $f(a), f(a + h)$, and the difference quotient $$\frac {f(a + h) − f(a)}{h},$$ where $h\neq 0$ and $f(x) = 2-6x+4x^2$. My work: $$f(a) = 2-6a+4a^2,\ \ f(a+h) = 2-6(a+h)+4(a+h)^2.$$ ...
-1
votes
3answers
44 views

Solve for $y$: $\frac{y+1}{y-1} = 10^{x^2}$ [on hold]

Can someone please show me the steps (all of them… yeah, even the obvious ones) to go from $$\begin{align}\frac{y+1}{y-1} = 10^{x^2}\end{align}$$ to ...
0
votes
0answers
15 views

Interpretation of Rates of Change

Suppose $C(t)$ represents the number of cars arriving at a particular station. I am looking for possible interpretation of the following: (i) $C(t)\Delta t$ (ii)$\frac{C(t+\Delta t) - C(t)}{C(t)\Delta ...
13
votes
4answers
150 views

Product of cosines: $ \prod_{r=1}^{7} \cos \frac{r\pi}{15} $

Evaluate $$ \prod_{r=1}^{7} \cos {\dfrac{r\pi}{15}} $$ I tried trigonometric identities of product of cosines, i.e, $$\cos\text{A}\cdot\cos\text{B} = \dfrac{1}{2}[ \cos(A+B)+\cos(A-B)] ...
3
votes
2answers
27 views

Number of ways to select subsets

In how many ways can two distinct subsets of the set $\text{A}$ of $k$ $(k \geq 3)$ elements be selected so that they have exactly two common elements? I started by choosing two elements (that ...
0
votes
1answer
34 views

What does it mean for a “formula to be undefined”?

I was covering the techniques used sketch rational functions of five different types as follows: However, then I encountered this: And, Ij just can't find out what it means for the formula to ...
6
votes
6answers
548 views

Systematically guessing integer roots of a cubic polynomial

Suppose I have a cubic equation, such as $$15x^3-4x^2-25x+14=0.$$ By the Hit and Trial method I know that one of the roots is $x=1,$ and hence I can solve the cubic equation with ease, as it will ...
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votes
0answers
19 views

Find the population [on hold]

Every year, the emigration rate from country A to B is 𝛼 (0 < 𝛼 < 1), whereas the emigration rate from country B to A is 𝛽 (0 < 𝛽 < 1). Note that the fluctuation in population of both ...
2
votes
3answers
324 views

Finding the roots of a different Quadratic equation from the roots of a Given Quadratic equation

The Question: If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$... Then find the roots of the equation $ax^2-bx(x-1)+c(x-1)^2=0$ My Attempt: The new equation can be ...
4
votes
3answers
54 views

Square roots equations

I had to solve this problem: $$\sqrt{x} + \sqrt{x-36} = 2$$ So I rearranged the equation this way: $$\sqrt{x-36} = 2 - \sqrt{x}$$ Then I squared both sides to get: $$x-36 = 4 - 4\sqrt{x} + x$$ Then I ...
3
votes
1answer
34 views

Find the inverse of the function given:

$$f(x)= \frac{5x}{(x − 2)}$$ My work: $$y=\frac{5x}{(x-2)}$$ $$x=\frac{5y}{(y-2)}$$ $$x(y-2)=5y$$ $$xy-2x=5y$$ $$\frac{xy-2x}{5}=y$$ $$f(x)=\frac{(xy-2x)}{5}$$ Any help is appreciated.
0
votes
2answers
28 views

If the i-th, j-th, and k-th terms in an AP are in a GP with ratio r, find $r$ in terms of $i, j$, and $k$

If the i-th, j-th, and k-th terms in an arithmetic progression are in a geometric progression with ratio r, find r in terms of i, j, and k. This is my result: (1) if $ik \ne j^2$ then ...
2
votes
4answers
70 views

According to Stewart Calculus Early Transcendentals 5th Edition on page 140, in example 5, how does he simplify this problem?

In Stewart's Calculus: Early Transcendentals 5th Edition on page 140, in example 5, how does $$\lim\limits_{x \to \infty} \frac{\dfrac{1}{x}}{\dfrac{\sqrt{x^2 + 1} + x}{x}}$$ simplify to ...
0
votes
0answers
34 views

Can This Expression Be Simplified? (Involves Square Roots)

I started with the expression $$ \frac{4mlt(1-\sqrt{1-\frac{v^2}{c^2}})c^2}{\sqrt{1-\frac{v^2}{c^2}}} $$ and have ended up at: $$ \frac{4mlt(c^2 - c \sqrt{c^2-v^2})}{\sqrt{1-\frac{v^2}{c^2}}} $$ ...
1
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3answers
33 views

What is the unknown angle?

So first off I started with the pythagorean theorem to find the missing leg of the triangle. \begin{align*} 5^2 + b^2 ={}& 8^2 \\ 25 + b^2 ={}& 64 \\ 64 - 25 ={}& 39 \\ \text{missing ...
1
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2answers
35 views

Representation in the complex plane.

Determine the set of representations in the complex plane for which: (a) $\frac{z-1}{z+1}$ is a real number; (b) $\frac{z-1}{z+1}$ is a pure imaginary number.
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1answer
46 views

Polynomials and geometric progression? [on hold]

Dividing $P(x)=a_5x^5 + 2x^4 + a_4x^3 + 8x^2 - 32x + a_3$ by $x-1$, we get $Q(x)=b_4x^4 + b_3x^3 + b_2x^2 + b_1x + b_0$, and $6$ as the remainder. Knowing $(b_4,b_3,b_2,b_1)$ is a geometric ...
1
vote
1answer
33 views

Question on Arithmetic and Geometric Progression

Problem: $S_1, S_2, S_3, \ldots ,S_n$ are the sum of $n$ terms of $n$ GPs whose first term is 1 in each case. However, the common ratios $r$ are $1,2, 3, \ldots,n$ respectively. Prove that ...
0
votes
1answer
56 views

How do I prove this nice inequality $x+3\sqrt[3]{xy^2}\geq4\sqrt{xy} $?

Let $x,y\geq0$. Prove that: $$ x+3\sqrt[3]{xy^2}\geq4\sqrt{xy} $$ Note: It's seems easy but when I tried to show it I went to complicated formula.
3
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4answers
129 views

Arithmetic and Geometric Progression Question 1

Problem: The second, third and sixth terms of an arithmetic progression are consecutive terms of a geometric progression. Find the common ratio of the geometric progression. My attempt: I ...
4
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2answers
83 views

Show that in any triangle, we have $\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$

Show that in any triangle, we have $$\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$$ where $R$ is the circumradius of the triangle. Here is my work: ...
2
votes
1answer
58 views

Find 4 numbers which create a ratio

Find four numbers which create ratio if its known that sum of first and last is equal to 14, sum of middle two is equal to 11 and sum of squares of all numbers is equal to 221 I got only that sum of ...
2
votes
3answers
64 views

How can $f(x)=x^4$ have a global minimum at $x=0$ but $f''(0)=0$?

$f(x)=x^4$ has a global minimum in $\Bbb R$ at the point $x=0$, but $f''(0)=0$. This case confuses me. For every $0\neq x\in I$, $f(x)>f(0)$. So how can it be that $f''(0)=0$, following $f'(x)$ ...
2
votes
3answers
55 views

Is there a simple way to define the $n$-th roots of the unity?

Is there a simple way to calculate the $n$-th roots of the unity? I gotta solve the equation $$\frac{z+1}{z-1}=\sqrt[n]{1}.$$
1
vote
3answers
222 views

A confusion in a calculation with complex numbers

Consider the followings: $$ 1+e^{ix}+e^{2ix}+e^{3ix}= \dfrac{1-e^{4ix}}{1-e^{ix}} $$ Then, we take absolute square to the both sides $$ |1+e^{ix}+e^{2ix}+e^{3ix}|^{2}= \dfrac{1-\cos4x}{1-\cos x} $$ ...
0
votes
1answer
25 views

How do I solve a polynomial equation with a square root variant?

Here is the equation: http://i.stack.imgur.com/UARLE.jpg I notice that the quadratic formula has been used to get the final answer. But something does not seem right because this the equation which I ...
1
vote
2answers
43 views

Fraction with numbers [on hold]

In an audience of people watching a film, 1/10 are under 16 years old, 3/5 are between 16 and 40 years old and the rest are over 40 years old. What fraction are over 40 years old? How to get the ...
0
votes
0answers
26 views

Properties of exponentiation proof

I'm trying to prove the following: "Let $x, y$ be non-zero rational numbers, and let $n,m$ be integers. Then we have $x^n x^m = x^{n+m}$." I've managed to prove by induction the case $n,m \geq 0$ ...
4
votes
1answer
39 views

Roots of $p(x)=\prod_{i=1}^{2n}(x-d_i)+k^2, \ \ \ \ n\in\mathbb N,\ k\in\mathbb R$

Let $$p(x)=\prod_{i=1}^{2n}(x-d_i)+k^2, \ \ \ \ n\in\mathbb N,\ k\in\mathbb R$$ where $d_i>0$ for all $i=1,\dots,2n$. Can I infer that $$p(x)=0$$ has only roots with positive real part?
0
votes
1answer
35 views

Prove base times altitude is a constant without resorting to area.

The area of a triangle is one-half base times altitude. This implies that, for $\bigtriangleup ABC$, $ah_A = bh_B=ch_C$, where $h_A$ is the length of the altitude dropped from point A to side BC, etc. ...
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votes
1answer
43 views

Select group of real numbers $x$ [on hold]

Select group of real numbers $x$, satisfy the inequality $$\frac{4x^2}{(1-\sqrt{2x+1})^2}< 2x+9$$ help guys!!
0
votes
2answers
21 views

Confused about proof of division

I thought I was familiar with the regular euclidian algorithm, but I am having trouble understanding a step in this proof from my notes, I am looking for any clarification. $\mathbf{Thereom:}$ Let ...
0
votes
1answer
27 views

Looking to understand proposition related to the fundamental theorem of algebra

I am having some problem understanding exactly what the following proposition is saying. Also, is this result have a common name? How important it is, etc. It is $\mathbf{Proposition:}$ Let ...
0
votes
2answers
50 views

Find six triples of positive integers $(a, b, c)$ such that in $ \frac{9}{a} + \frac{a}{b} + \frac{b}{9} = c$.

Solve for $a, b$ and $c$ in the following equation such that Find six triples of positive integers (a, b, c) such that $$ \frac{9}{a} + \frac{a}{b} + \frac{b}{9} = c$$ I have tried various ...
-2
votes
1answer
45 views

Solve for b and d

Solve for b and d in the following equation. A triangle with sides $(a, a, b)$ has the same area and the same perimeter as a triangle with sides $(c, c, d)$ where $a, b, c$ and $d$ are positive ...
2
votes
0answers
52 views

What is the maximum amount of solutions to $f(x+1)f(x)= ax^2+bx+c$.

$f(x)$ is going to be in the form $mx+h$ thus, $(mx+m+h)(mx+h) = ax^2+bx+c$. With basic algebra $m= \pm \sqrt{a}$. Also $(m+h)(h)=c$. I would guess that because $(m+h)h=c$ has two solutions max if $m$ ...
2
votes
5answers
68 views

Can't determine if given relation is equivalence relation

Definition of relation ~ $(a,b)$ ~ $(c,d)$ $\iff$ $bc^2=da^2$, where $(a,b),(c,d)$ are from $\mathbb{R}\times\mathbb{R}$ and $(a,b),(c,d)$ are different from $(0,0)$ First of all, I wonder if ...