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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2answers
179 views

Express in the form (x-a)

How to express $2x-1$ in the form $x-a$ ? Isn't $x-1/2$ wrong? And how to express $2x+3$ in the form x-a? this is for using it in the remainder theorem: when $f(x)$ is divided by $x-a$, the ...
2
votes
2answers
805 views

Finding the intersection of a two points and an arbitrary axis

Given two points I would like to find where the line joining them intersects an arbitrary axis. For example, if I had one point (5, 10) and another at (50, 100) I can be sure that somewhere a line ...
0
votes
1answer
480 views

Vector proof for midpoints of 2 sides and diagonal intersection

Point $A, B, C, D$ have position vectors $a, b, c, d$ respectively relative to an origin O. If $P$ divides $AB$ in the ratio $1:2$ and $Q$ divides $CD$ in the ratio $1:2$, obtain an ...
4
votes
1answer
172 views

How can I calculate what my credit card balance will be next month?

If my current credit card balance in July is \$1,000 USD, my credit card's APY is 20% and this month I made a payment for \$100 on time (to avoid late fees)... What will my balance be in August? I ...
4
votes
5answers
408 views

Showing $a^2 < b^2$, if $0 < a < b$

Lately, I've been stumbling with proofs of inequalities. For example: Given $0 < a < b$ Show $a^2 < b^2$ The only thing I've been able to come up with so far: $a^2 < b^2$ ...
0
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2answers
241 views

Why people have to find quadratic formula,isn't that the formula cannot solve a polynomial with 2 and 1/2 degree?

Why people have to find quadratic formula,isn't that the formula cannot solve a polynomial with 2 and 1/2 degree? and just curious, how many roots does a polynomial with 2 and 1/2 degree have and how ...
2
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1answer
130 views

Vector linear combination problem

Point $A$ and $B$ have position vectors $\vec a$ and $\vec b$ respectively relative to an orgin $O$. The point $D$ is such that $\overrightarrow{OD} = k\overrightarrow{OA}$ and the point ...
1
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3answers
177 views

A nonlinear single variable equation

Say I have $f(k)$ for all $k = 1, \ldots, 2^n$, and $0\le f(k)\le n$. I would like to solve for $x$: $$ x\sum _{k=1}^{2^n }\frac{1}{f(k)+x} =1. $$ And also this if possible, $$\sum ...
3
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2answers
129 views

How do I prove that $\sum \limits_{i=1}^n{p_i\left(x_i-\bar{x}\right)^2} = \frac{1}{2} \sum \limits_{i, j=1}^n {p_ip_j\left(x_i-x_j\right)}^2$?

How can I show if $\displaystyle\bar{x}=\sum_{i=1}^n{x_ip_i}$ , then $$\sum_{i=1}^n{p_i\left(x_i-\bar{x}\right)^2}=\frac{1}{2}\sum_{i=1}^n\sum_{j=1}^n{p_ip_j\left(x_i-x_j\right)}^2$$ is true? (This ...
1
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1answer
313 views

Just wondering what this imo problem is asking and how to solve

Just wondering what this imo problem is asking(it looks simple but i don't understand what's important in the question) and how to solve: Suppose that $s_1,s_2,s_3,\ldots$ is a strictly increasing ...
0
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1answer
398 views

Express in terms of single vector

Can you guys give me a hint for this problem. ABCDEF is a regular hexagon. Express in terms of a single vector the sum of the vectors, $4\overrightarrow{AB}$, $2\overrightarrow{AC}$, ...
2
votes
3answers
91 views

Solving for the interval of time during which the height of a thrown ball is at least “h” feet

In dealing with inequalities I've run into a certain peculiarity which I am currently unable to explain. The example: Find the interval of time during which the ball is at least 32 feet above ground. ...
2
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3answers
917 views

How to solve a polynomial inequality?

Ok guys, I need some more help with a question for my girlfriend. Basically she was given a problem on a test/quiz and the only way I know how to do it is with a method that she hasnt learned in class ...
5
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6answers
9k views

Solving inequalities with “x” in the denominator

Solving inequalities with "x" in the denominator has always been a stumbling block for me. Other than understanding how a particular expression, such as 1/x, works (in this case, x cannot be zero), ...
6
votes
6answers
11k views

How to solve this equation with two square root terms?

So guys, my girlfriend is taking a college algebra class this summer and I figured I would help her study for her upcoming final because I am an engineering major and this kind of math would be easy ...
2
votes
2answers
472 views

Types of divergence

My teacher said there are two main ways a sequence can diverge, it can increase in magnitude without bound, or it can fail to resolve to any one limit. But maybe that second kind of divergence is too ...
2
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2answers
133 views

$\sqrt[x]{y} = y^{1 / x}$? Need help with algebra of exponentiation

I need help with algebra of exponentiation. $$\begin{align*} \sqrt{x^2} &= (x^2)^{1/2} &\qquad &\text{(since }\sqrt[x]{y}=y^{1/x}\text{)}\\ &= x^{2(1/2)} &&\text{(since ...
4
votes
1answer
289 views

Domain of a function

I am confused about this problem: Find the domain of the function, $$f(x)=\frac{x^3-1}{2x^2+5}.$$ I'm guessing it's all real numbers but the book gives a different answer. The book gave ...
9
votes
2answers
9k views

How to calculate percentage of value inside arbitrary range?

So pardon if this is a simple question... I have a slider that returns a value in a given range, so: min: 174 max: 424 slider current value: 230 I want to treat ...
3
votes
3answers
1k views

How to add compound fractions?

How to add two compound fractions with fractions in numerator like this one: $$\frac{\ \frac{1}{x}\ }{2} + \frac{\ \frac{2}{3x}\ }{x}$$ or fractions with fractions in denominator like this one: ...
3
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2answers
2k views

Solving system of equations which contain sin and cos

I require some help to push me in the right direction to solve these equations. $$t_1 = P_1\sin(A)\sin(B) + P_2\cos(A)\cos(B)$$ $$t_2 = P_3\cos(A)\sin(B) + P_4\sin(A)\cos(B)$$ where $t_1, t_2, P_1, ...
2
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2answers
132 views

Converting Single Equation To A System Of Equations

Assuming $f\neq 0$ I'm trying to rewrite $$0 = 2 \cdot g \cdot ((x-a)^2 + 1) - 2 \cdot d \cdot f \cdot (x - a)$$ into a system of equations of the form $a = $(something not containing $d$) $d = ...
2
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2answers
219 views

Algebraic word problem question

Since the beginning of algebra, I have always been terrible at word problems. I don't know why. I have this word problem here and would love for someone to explain in large depth how to solve these ...
1
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2answers
747 views

Angle between 2 faces of tetrahedron

Two faces ABC and DBC of a tetrahedron ABCD are right-angled triangles with $\angle ACB = \angle DCB = 90^{\circ}$. Given that the edge DA is perpendicular to the face $ABC$, $\angle CBD = ...
0
votes
2answers
109 views

Simple algebra question

I'm working on a problem for my accounting class and I'm trying to figure why 0.10 would become 1.1 on the other side of the equation. Math: ...
10
votes
1answer
572 views

Prove that minimum of $\lambda \sin \theta + (1 - \lambda) \cos \theta \le -\dfrac{1}{\sqrt 2}$

I need a little nudge to the finish for the last bit of this problem. Express $\lambda \sin \theta + (1 - \lambda) \cos \theta$ in the form $R \sin (\theta + \phi)$, where $R(R>0)$ and $\tan ...
2
votes
1answer
161 views

Need help in deriving condition for quartic to have only one double root

Given a polynomial of degree four: $ax^4+bx^3+cx^2+dx+e$, with $a,b,c,d,e$ real and $a\neq 0$, how do I derive the condition for there to be exactly distinct 3 real roots (i.e., one root is repeated)? ...
1
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1answer
118 views

$(a_{1}+ a_{2} + …+a_{k})^{n}$ where $k >2$, what does it generate?

Binomial expansion generates the Pascal triangle but what does it generate when you have different amount of terms there? You can see here the geometric generation with only 2 terms. I am interested ...
5
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4answers
232 views

Evaluate $\int{\frac{4x}{(x^2-1)(x-1)}dx}$

So I know I need to use the partial fractions method to solve this integral. However when I split it as: $$\frac{4x}{(x^2-1)(x-1)} = \frac{Ax + B}{x^2-1} + \frac{C}{x-1}$$ I find that I can't solve ...
7
votes
2answers
180 views

Notation of Elementary Inverse Functions

Let $y = f(x) = \sqrt{2x + 1}$ for $x \geq -1/2$. Then, $f$ is injective on its domain and therefore its inverse is well-defined. To find the inverse, we simply invoke the necessary algebraic ...
7
votes
7answers
462 views

Trigonometric equality: $\frac{1 + \sin A - \cos A}{1 + \sin A + \cos A} = \tan \frac{A}{2}$

Can you guys give me a hint on how to proceed with proving this trigonometric equality? I have a feeling I need to use the half angle identity for $\tan \frac{\theta}{2}$. The stuff I have tried so ...
1
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1answer
322 views

Find number of interesting numbers (China TST 2011)

A positive integer $n$ is known as an interesting number if $n$ satisfies $$ \left\{\frac{n}{10^k}\right\} > \frac{n}{10^{10}} $$ for all $k=1, 2, \ldots, 9$, where $\{x\}=x - \lfloor x \rfloor$. ...
6
votes
3answers
241 views

Intuition behind problem in (classical) algebra

To give some background, the question is to show that if $a=b+c$ then $$a^4+b^4+c^4 = 2a^2b^2+2b^2c^2+2c^2a^2$$ Which, for completeness, I was able to do by squaring twice $$(a-b-c)^2=0$$ gives ...
3
votes
0answers
69 views

Uses for the generalised f-mean, functions with larger/smaller f-means

What are some uses of the generalized f-mean outside of the geometric mean and the power means? Also, is there a known way to compare two functions and find out which will yield a larger f-mean (ex: ...
1
vote
2answers
42 views

Increasing rectangular surface

I have this problem: One side of a rectangular is lengthen by 40%. How much the other side has to be lengthen in order the surface to be bigger by 47%. The solution is given to be "by 5%". Probably ...
2
votes
2answers
86 views

Which steps I have to do to get this equation?

I don't know what to do to derive the right side from the left side: $$\frac{B}{1+r} = B - \frac{r B}{1+r}.$$
2
votes
1answer
165 views

Logarithm rules

What can I do with these expression: $2^{\log _{\frac{4}{3}}n}$ and $2^{\log _{4}n}$ if I don't want to have $n$ in the exponent? I tried nothing because I didn't have any good ideas. Thanks.
7
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3answers
1k views

Are there numerical algorithms for Roman numerals?

In positional number systems there are algorithms for performing certain operations, like long division, to name one of the simplest. This works for positional systems, whatever base. I realize in ...
2
votes
2answers
1k views

Find $\sin \theta$ and $\cos \theta$ given $\tan 2\theta$

Can you guys help with verifying my work for this problem. My answers don't match the given answers. Given $\tan 2\theta = -\dfrac{-24}{7}$, where $\theta$ is an acute angle, find $\sin \theta$ ...
3
votes
3answers
780 views

A hyperbola as a constant difference of distances

I understand that a hyperbola can be defined as the locus of all points on a plane such that the absolute value of the difference between the distance to the foci is $2a$, the distance between the two ...
2
votes
1answer
299 views

Zooming formula

I am sorry if this is a noob question, I need help with relatively simple math problem and assurance that I understand the problem correctly. I have a map-like program that zooms in and zooms out if ...
9
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2answers
709 views

Motivating algebra from quadratic equations

This question gave me pause for thought. We have a quadratic equation $ax^2+bx+c=0$. How much algebra can be motivated from the standard solution. Comments point out that the formula does not apply in ...
123
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17answers
12k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
26
votes
10answers
3k views

$i^2$ why is it $-1$ when you can show it is $1$? [duplicate]

We know $$i^2=-1 $$then why does this happen? $$ i^2 = \sqrt{-1}\times\sqrt{-1} $$ $$ =\sqrt{-1\times-1} $$ $$ =\sqrt{1} $$ $$ = 1 $$ EDIT: I see this has been dealt with before but at least with ...
1
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2answers
83 views

Transforming a sequence using polynomials

Let $$f(x) = \sum_{n \geq 0} a_n x^n = \frac{x-2x^3}{4x^4 - 5x^2 + 1}$$ Now I need to identify a "concrete formula" for $a_n$. This should be done by using the following proposition: For a ...
5
votes
3answers
369 views

What does the notation $\binom{n}{i}$ mean?

What do the parentheses next to the summation involving the binomial coefficients mean? Like this: $$\sum _{i=0}^{n} \binom{n}{i}a^{(n-i)}b^i=\left(a+b\right)^n $$
5
votes
1answer
443 views

Solving for the center of mass of a Semi Circle (without integration) [duplicate]

Possible Duplicate: Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$ For fun, I was trying to solve this problem without doing calculus. After dinking around with it for a while, I ...
4
votes
2answers
299 views

Find angle subtended by overlapping circles [duplicate]

Possible Duplicate: Solving $2x - \sin 2x = \pi/2$ for $0 < x < \pi/2$ I need a little nudge for this problem. I have figured out everything but the last bit(albeit the most ...
1
vote
2answers
233 views

Solving graph of trigonometric function and rational function

I am trying to work out how to solve the following functions graphically. On the same axes, draw the graph of $y = \sin x$ ($x$ in radians) and $y = \dfrac{1}{x}$ for values of $x$ between $0.5$ ...
3
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2answers
98 views

Help with simultaneous equation with additional term

I hoped someone can help me with 3 simultaneous equations with an additional condition. I can easily solve the following 3 equations using substitution in terms of $S_1$, $S_2$ and $S_3$" ...