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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1answer
21 views

Equality of polynomials and their equivalent fraction forms

A polynomial of the form $\frac{p(t)\cdot (x+1)}{(x+1)}$ is obviously equal to $p(t)$ because the binomials cancel out, where $p(t)$ is just any arbitrary polynomial. But the first form is undefined ...
4
votes
2answers
68 views

Intersection of semicircle and parabola (Omar Khayyam)

(Source: The History of Mathematics 7th Edition, David Burton) I can't even get through question a.. Could someone give a hint? The only thing I can think of is the Pythagorean Theorem, but it ...
0
votes
1answer
21 views

How to find the resultant vector in a word problem

If the wind blows at 20 mph due West and an airplane heads South at 400 mph, what is the resultant speed and direction of the plane?
1
vote
7answers
134 views

How do I solve the equation $e^{\ln(2x+1)} = 5x$?

The problem is $$e^{\ln(2x+1)} =5x$$ I've tried using natural logs to both sides like.. $2x+1= \ln 5x $ But I'm not sure if $\ln$ and $e^{\ln}$ cancel out.
1
vote
2answers
39 views

$x^2 + (k-3)x + k = 0$, ranges of k for roots to be of same sign

I need some help on the following. The quadratic that I am dealing with is $x^2 + (k-3)x + k = 0$, and I need to find ranges of values of $k$, for which the roots will have the same sign. For the ...
2
votes
3answers
47 views

Is this the correct period?

What is the period for the following: $$ y = 10 \sin\Bigl(\frac{2\pi}{365}(x-50)\Bigr) $$ Is the period $$ \frac{2\pi}{\frac{2\pi}{365}} $$ which would be $365$?
0
votes
1answer
52 views

How do I use the bowtie method to multiply $(2x-27)(-x+15)$?

The bowtie method seems like an easy concept to have down, but how is it used to multiply binomials such as $(2x-27)(-x+15)$? Calculating the answer is not the problem, because I can get ...
1
vote
2answers
64 views

How many such polynomial exist?

Find the number of second-degree polynomials $f(x)$ with integer coefficients and integer zeros for which $f(0)=2010$. I got: $$P(x) = ax^2 + bx + c \implies P(0) = c = 2010$$ Let $P(r_1, r_2) ...
2
votes
4answers
44 views

How to find if a point is outside a circle circumference area?

I'd like to know if it's possible to calculate if a point is inside or outside the circle circumference area based on it's $x$ and $y$ values ? Example, $(x, y)= (0.85, -0.9)$ and the radius is $1$
0
votes
2answers
32 views

Find the greatest value of $b$.

If one of the roots of the equation $(a-b)x^2+ax+1=0$ is double of the other and is real , find the greatest value of $b$. $\color{green}{a.)\ \dfrac98} \quad \quad \quad \quad \quad b.)\ ...
0
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0answers
17 views

Cubic formula derivation

François Viète used the substitution $y=z-\frac{p}{3z}$ to transform the depressed cubic $y^3+py=q$ into an 'almost' quadratic equation. ...
-2
votes
2answers
50 views

Panicking because I forgot how to do basic algebra? [closed]

How do I convert 6p / 2py - 4p to simplest form? It's not working for me..
2
votes
0answers
39 views

Is this system of inequalities (and equality) tractable?

I have some real parameters here. The $\mu_i$ - for $i=1,2,3,4,5$ - are 'convex coefficents' in that $\mu_i\geq 0$ and $\sum_{i}\mu_i=1$. The $x$ and $z$ are such that $x^2+z^2\leq 1$. The ...
1
vote
1answer
44 views

Probability of not making a shoe pair.

Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The ...
2
votes
5answers
81 views

If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ then..

If $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ then (A) $a=c$ (B) either $a=c$ or $a+b+c+d=0$ (C) $a+b+c+d=0$ (D) $a=c$ and $b=d$ I solved $\frac{a+b}{b+c}=\frac{c+d}{d+a}$ and got $a(a+b+d)=c(c+b+d)$ and ...
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votes
1answer
40 views

What is X if position value is between 0 and 500 and I need it has -1 to 1?

Let's say I have a line from 0 to 500, and I'd like to know in which position a point is from -1 to 1, where centre is 250, how can I do that ? Any suggestions ? Thanks! So far this is what I've ...
3
votes
1answer
65 views

Why doesnt this Combinatoric work two ways?

There are two distinguishable flagpoles, and there are $19$ flags, of which $10$ are identical blue flags, and $9$ are identical green flags. Let $N$ be the number of distinguishable arrangements ...
-4
votes
3answers
61 views

mental ability math question [closed]

A worker may claim Rs. 15 for each km if he travels by taxi and Rs. 5 for each km if he drives his own car. If in one week he claimed Rs. 500 for travelling 80 km, how many kms did he travel by taxi ? ...
2
votes
1answer
26 views

On square roots and Geometric means

Let $a$ and $b$ be two negative integers, say $-4$ and $-1$ So $$\sqrt{ab}=\sqrt{-4\cdot(-1)}= + 2 $$ But then again, $\sqrt{ab}$ is the geometric mean between $a$ and $b$, so it is supposed to ...
-1
votes
1answer
23 views

I have a contrast enhancement function for pixel intensities between 0-1. How can I reverse this

This is the function to enhance contrast for pixel intensities between $0-1$ $f(x) = \cfrac{1}{1 + \mathrm{e}^{\text{gain}(\text{cutoff}-x)}}$ I have a whole bunch of images that I need to reverse ...
6
votes
6answers
473 views

Why does fixed point iteration only produce the solution greater than $1$ to the equation $Mx = e^x$ for $x \in \Bbb R$?

The equation $Mx = e^x$, when $M > 0$. I know that the first solution must be at the tangent where the line $Mx$ crosses $e^x$, so $M$x has gradient $e^x$. This leads to $x(e^x) = e^x$, $x = 1$ ...
2
votes
4answers
99 views

Find the principal solutions of the trigonometric equation $\cos x-\sin x+\sin 2x+3\cos2x+1=0$

I am unable to simplify the expression. If I simplify the double angles, it leaves me with a nasty expression, $\cos x-\sin x+2\sin x\cos x+6\cos^2 x-2=0$. What do I do next. Some hints, please. ...
4
votes
3answers
44 views

Inequalities and rearrangements

Some children are arranged in two rows, so that each child in the front row is taller than the child behind him in the back row. The children are now rearranged in increasing order in each row. Show ...
-3
votes
2answers
59 views

Use substitution $x=a\sec(\theta)$ to show that [closed]

Use substitution $x=a\sec(\theta)$ to show that $$\displaystyle\int_{a\sqrt 2}^{2a} \frac{dx}{x^3\sqrt{x^2-a^2}} = \frac{3 \sqrt 3 + \pi - 6}{24 a^3}$$ Need help! Could someone show me the working ...
0
votes
4answers
59 views

Given $x^2 + y^2 = 34xy$, show that $\log\left(\frac{x+y}{6}\right)= \frac{\log x + \log y}{2}$

If $x^2 + y^2 = 34xy$, show that $$\log\left(\frac{x+y}6\right)= \frac{\log x + \log y}{2}.$$ I tried to put log into the first equation, but I have no idea about how the $34$ being simplified in the ...
0
votes
2answers
31 views

Simplification of algebraic expression

I have got a big algebraic expression for function $\vec{a} = \vec{a}(s)$, that i simplified to the form $$\frac{d \vec{a}}{ds} + \frac{a - \sin a}{a^3} (\vec{a} (\vec{a} \cdot \frac{d \vec{a}}{ds}) - ...
0
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2answers
35 views

Average speed problem

"I take a journey and, due to heavy traffic, crawl along the first half of the complete distance of my journey at an average speed of $10$ mph. How fast would I have to travel over the second half of ...
0
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0answers
50 views

Can you help me with creating a formula [closed]

I make aluminium windows and doors and I would like to create a formula based on an overall width. The scenario is: There are 2 fixed panels of equal length. What are their lengths' as an equation? ...
1
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2answers
28 views

How to solve for $t$ in the equation $m=-70/(t-25)$?

In the equation $$m= -\frac{70}{t-25}$$ How would I solve the equation for $t$ so that I get t equal to something $m$? I've tried multiplying both sides by $t-25$, but that just leaves me with ...
2
votes
3answers
36 views

Values of $p$ for which quadratic possess at least one positive root.

For what values of $p$ would the equation $x^2+2(p-1)x+(p+5)=0,\ \ \{x,p\}\in \mathbb{R}$ possess at least one positive root ? I tried $$[2(p-1)]^{2}-4(p+5)\geq 0\\~\\ \implies p\geq 4 \cup p\leq ...
1
vote
3answers
74 views

simplify and evaluate $\frac{\tan80^\circ-\tan20^\circ}{1+\tan80^\circ\tan20^\circ}$ [closed]

How do you simplify and evaluate $\dfrac{\tan80^\circ-\tan20^\circ}{1+\tan80^\circ\tan20^\circ}$? What is the problem asking?
-7
votes
3answers
57 views

Find exact values of $\tan(105^\circ)$ and $\tan(11\pi/12)$ without calculator [closed]

How do you find the exact values of the following without using a calculator? $$\tan(105^\circ) \qquad \tan(11\pi/12)$$
0
votes
6answers
102 views

Factoring $ x^2 + x +1 > 0$ from Spivak Calculus exercise

Hi!! I found me in trouble when I saw the solution of a simple inequality, that can be found at the end of the first chapter, that is the exercise 4 - (viii): $x^2+x+1 > 0$. Very easy to solve I ...
1
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3answers
32 views

Area/Sector of a circle: A cow is tethered by a 100-ft rope to the inside corner of an L-shaped building

A cow is tethered by a $100ft$ rope to the inside corner of an L-shaped building, as shown in the figure. Find the area that the cow can graze. (Let $a = 30 ft$, $b = 60 ft$, $c = 100 ft$, $d = 70 ...
2
votes
3answers
73 views

When is $\tan(a+b)$ undefined? [closed]

For what values of $a$ and $b$ is $\tan(a+b)$ undefined? What is the relationship between $a$ and $b$ when it is undefined? What about for $\tan(a-b)$?
1
vote
3answers
58 views

How many possible guesses?

A game show offers a contestant three prizes A, B and C, each of which is worth a whole number of dollars from $ 1$ to $ 9999$ inclusive. The contestant wins the prizes by correctly guessing the ...
0
votes
2answers
35 views

Co-ordinate geometry involving straight lines $7x-y-32 = 0$ and $3y-2x+1=0$.

Let $P$ be the point of intersection of the lines $7x-y-32=0$ and $3y-2x+1=0$. Lines are drawn through $P$ making intercepts of equal magnitude on the co-ordinate axes. Find the equation of these ...
2
votes
1answer
30 views

Find $p$ and $q$ in $x^2-px+q=0$

If $p$ and $q$ are the roots of the equation $x^2-px+q=0,\ \{x,p,q\}\in\mathbb{R} $, then find $p$ and $q$. I tried sum and product of the roots formula and got , $$p+q=p \\pq=q$$ I found $q=0$ ...
5
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1answer
65 views

How many ways are there to shake hands?

In a group of $9$ people, each person shakes hands with exactly $2$ of the other people from the group. Let $X$ be the number of possible ways to perform these handshakes. Take $2$ handshake ...
-3
votes
1answer
30 views

Dividing a line segment based on a ratio [closed]

What is the standard form of the equation for this circle? (A) $(x – 4)^2 + (y + 5)^2 = 5.5$ (B) $(x – 4)^2 + (y + 5)^2 = 30.25$ (C) $(x + 4)^2 + (y – 5)^2 = 30.25$ (D) $(x + 4)^2 + (y + 5)^2 = ...
3
votes
4answers
109 views

Find the matrix $\mathbf{A}$ if $A\binom{7}{-1} = \binom{6}{2}.$

Find the $2\times2$ matrix $A$ where $A^2=A$ and $$A\begin{pmatrix} 7 \\ -1 \end{pmatrix} = \begin{pmatrix} 6 \\ 2 \end{pmatrix}.$$ I tried plugging in: $A= ...
0
votes
3answers
55 views

Let $x_n>a$ for all indices $n$, and $x_n \rightarrow b$. Prove $b \ge a$.

My question then is: Suppose $x_n>a$ for all indices $n$, and $lim x_n=b$. Prove that $b \ge a$. My attempt: I am not going through contradiction-in-conclusion method (i.e. suppose $b<a$ since ...
0
votes
1answer
41 views

Different answers of a quadratic equation.

given $4x^2−4x-5=0$ we all know the solution but what my teacher showed me is different after we get the \begin{align*} x & = \frac{4 \pm \sqrt{96}}{8}\\ x & = \frac{4 \pm \sqrt{4 \cdot ...
1
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4answers
132 views

Is 4th root of $-1$ the same as $i^2$?

I am using mathway to check my algebra problems and tried entering in the 4th root of $-1$ (or $(-1)^{1/4}$. I get the same term back, but I thought that since the square root of $-1$ is $i$, that ...
-2
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0answers
58 views

Find the value of $ab$ [closed]

If $$a^6 -b^6=67658$$ Find the value of $ab$
6
votes
2answers
41 views

Sujection, finite set, $|X| \le n$? [closed]

Suppose that $\{1, 2, \dots, n\} \to X$ is a surjection. How do I show that $X$ is a finite set and that $|X| \le n$?
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votes
8answers
61 views

Find the smallest integer $n$ [closed]

Find the smallest integer $n$, such that $$n\left ( \sqrt{101}-10 \right )> 1$$
4
votes
2answers
100 views

how to rationalize $\frac {x-8}{\sqrt[3]{x}-2}$

In order to resolve a limit, I need to rationalize $\frac {x-8}{\sqrt[3]{x}-2}$. I tried multiplying it by $\sqrt[3]{x^3}$ or $\sqrt[3]{x^2}$ but with no much success. It seems that I can't use ...
-4
votes
1answer
41 views

How many miles did Jen run? [closed]

Jen runs twice as fast as her friend Amy. If Amy runs 3mph, how long will it take Jen to run 6 miles? 9 miles?
0
votes
0answers
38 views

Newton's Law of Cooling example: roasted turkey

Need help with part B. I keep getting negative answers. A roasted turkey is taken from an oven when its temperature has reached 185°F and is placed on a table in a room where the temperature is 66°F. ...