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
votes
1answer
80 views

Expand $(a-b)^3$ without formulas

When I solve $(a-b)^2$ I take $aa-ab-ab+bb$ I do not use formulas at all because I only forget them. To solve the above example all i do is to multiply one variable or constant at the time but when I ...
1
vote
1answer
43 views

The problem of surds and indices [closed]

If $$ \frac 12 \times \left(\sqrt[3] a + \sqrt[3] b + \sqrt[3] c\right) = \frac 1{\sqrt[3] 3 - 1}\;\;\;(a>b>c)$$ then what is $a - 2b - 3c $ ?
2
votes
1answer
40 views

Understanding Why Partial Fractions Works [duplicate]

I was wondering why, not how, partial fractions work the way we are normally taught to do. To be specific: We are told that, when we have a second degree expression on the bottom that can't be ...
1
vote
2answers
30 views

Given the volume of a stack of cubes, find the number of cubes

I was trying to solve a code challenge but could not wrap my head around the math. Your task is to construct a building which will be a pile of n cubes. The cube at the bottom will have a volume ...
3
votes
2answers
63 views

Sum of real values of $x$ satisfying the equation $(x^2-5x+5)^{x^2+4x-60}=1$

I have this equation from this paper (Q.63) Find the sum of all real values of $x$ satisfying the equation-$(x^2-5x+5)^{x^2+4x-60}=1$. My attempt- ...
1
vote
1answer
21 views

Find the positive difference of all possible values

This problem is quite challenging to me. It is highly appreciated if someone can help me with it or give me an hint. Thank you very much! Find the positive difference of all possible values of ...
0
votes
3answers
116 views

How to solve the equation $ x^{13} = 1$ by radicals?

Is there any elementary way to solve the equation $ x^{13}= 1 $ by means of radicals? If not, how to get all the solutions? Remark: The transcendental form of the solution by means of sines and ...
0
votes
1answer
50 views

Function Question: Where does the $3$ come from?

Can someone explain me how did ''3'' appear here ?
-1
votes
1answer
16 views

Working through summations to show equation

Given equation 1: $$E = \sum_{k=1}^N \tau x_k g(\frac{n_k}{\tau}) + \sum_{k=1}^N n_kh(\frac{n_k}{\tau})$$ equation 2: $$E = \frac{1}{2}\gamma X^2 + \epsilon \sum_{k=1}^N |n_k| ...
-1
votes
2answers
24 views

show equivalence of forumulas

Give Function A: $$ \frac{1}{2} \gamma X^2 - \frac{1}{2}\gamma \sum_{i=1}^N n_k^2 $$ and Function B: $$ \epsilon \sum_{i=1}^N |n_i| + \frac{\eta}{\tau} \sum_{i=1}^N {n_i}^2$$ Can you show that ...
0
votes
0answers
24 views

Showing Left Side to Right Side.

Let $\mathbf x$ is a $(p\times 1)$ vector, $\mathbf\mu_1$ is a $(p\times 1)$ vector, $\mathbf\mu_2$ is a $(p\times 1)$ vector, and $\Sigma$ is a $(p\times p)$ matrix. Now I have to show ...
2
votes
1answer
58 views

How to graph $\frac{(x+3)}{(x+1)}$?

I'm not looking for plotting values. I mean something which can be done without it. For example, for the function $f(x) = mx + b$, we can use the slope $m$ and the y-intercept $b$ to easily graph the ...
0
votes
3answers
44 views

What is the logic behind the method used to shift a graph horizontally? [duplicate]

I understand why we add or subtract from $f(x)$ to move the graph vertically. To move it up, we move all the values up by adding whatever constant value we want. However, when it comes to horizontal ...
0
votes
2answers
37 views

Finding Launch Angle for Two Projectile Collision

I'm trying to figure out the general equation for calculating the launch angle of one projectile required when trying to find a collision between that projectile and another. For the equation I've ...
-2
votes
0answers
30 views

How to get the value a in a expression that have log function? [closed]

There is an inequality ln(2a+1)>a Could someone show me step by step, how to get the range of a? Thanks~
3
votes
2answers
25 views

Precalc Optimization?

I need help with an optimization problem. I have a rectangle space being fenced. Three sides are fenced with a material costing 4 dollars and the last side costs 16 dollars. I was given that the area ...
4
votes
1answer
35 views

A projectile is launched with a velocity of 30 m/s at 40° to the ground. To two decimal places, what is its horizontal velocity in m/s?

A projectile is launched with a velocity of $30 \, m/s$ at $40^\circ$ to the ground. To two decimal places, what is its horizontal velocity in $m/s$? A ball is thrown downward from the top of a ...
2
votes
1answer
48 views

Inverse of almost quadratic functions

Hi all first question here, I'm currently trying to invert the following two (two since the $\pm$ sign) functions: $y = x^2 \pm \sqrt{(ax + b \sin{\theta})^2 + (b\cos{\theta})^2}$ Where a,b, ...
0
votes
3answers
28 views

shortcut technique to solve algebric problem

If we multiply three consecutive numbers, we get 120. what the summation of those numbers? Is there any shortcut way of solving this problem without doing much ...
5
votes
1answer
43 views

Prove : $2(x^3+y^3+z^3)+3xyz \ge 3(x^2y+y^2z+z^2x)$ with $x,y,z \gt 0$

At first I tried to divide both side by $xyz$, the inequality became: $$2\sum {\frac{x^2}{yz}}+3 \ge 3\sum{\frac xy}$$ Let $$\frac xy = a;\frac yz = b;\frac zx = c;$$ So all we have to prove is ...
4
votes
3answers
105 views

Why is $5\tan(54^\circ) = \sqrt{25 + 10\sqrt{5}}$ and $\tan\left(\frac{\pi}{5}\right) = \sqrt{5 - 2\sqrt{5}}$?

On the Wikipedia Page about Pentagons, I noticed a statement in their work saying that $\sqrt{25+10\sqrt{5}}=5\tan(54^{\circ})$ and $\sqrt{5-2\sqrt{5}}=\tan(\frac {\pi}{5})$ My question is: How would ...
1
vote
2answers
36 views

Dividing an integer into a fixed number of integers

What is the formula for dividing an integer into a fixed number of integers where the greatest distance between consecutive integers is 1. Dividing 10 into 4 integers we can get: ...
0
votes
3answers
114 views

Between which integers is $-3 + \sqrt 5$ situated? [closed]

I can't understand how to solve this question . I'll be thankful if you help me. Please solve it and post your complete solution & a little description of what you did. Between which integers ...
0
votes
2answers
62 views

When can I simplify an equation?

Suppose I have a few ecuations: $$\cos^2(x) = \sin(x)\cos(x) \Rightarrow cos(x) = sin(x) $$ $$ x^2 + 3x \ge 2x \Rightarrow x(x+3) \ge 2x \Rightarrow x+3 \ge 2$$ Which of them are true and ...
0
votes
2answers
89 views

Where would this be on a range? ($0$ to $100\%$) Algebra [closed]

Ok, we have two expressions: $30x + 1500$ $19x + 9000$ The first thing to note is, the first expression is $100\%$ and the second expression is $0\%$. The second thing is to note in all of ...
3
votes
5answers
85 views

Is the function $f(x) = x^2 - 2x$ one-to-one?

The answer given is that it's not because it's a parabola and hence, would fail the horizontal line test, i.e., two values of $x$ will have the same value of $f(x)$. However, how can we prove the ...
-2
votes
0answers
26 views

Exercise in relative frequency of buying a coffee

Pete, Douglas and Surya buy each other rounds of coffee when they are in the office. They want to do so at a frequency that is fair, proportionate to each individual’s attendance and the attendance ...
0
votes
0answers
37 views

Finding $x^2$ and $y^2$ of hyperbola

Currently, I am trying to the $x^2$ and $y^2$ of a hyperbola. I have the vertices at $(-1, -1)$ $(5, -1)$ I have the focus at $(-4, -1)$ $(8, -1)$ I know that the distance between two vertices ...
-2
votes
3answers
92 views

Prove $\sqrt{x^{2} + y^{2}} \le |x| + |y|$ [closed]

How do I prove this? x and y are real numbers. Thanks for the help.
0
votes
0answers
31 views

Trouble with this simplification

I am following a proof which makes what seems to be a "big" jump. From $$ \lim_{x \to \infty} \frac{\mu \{1 - \Phi(x)\} + \sigma \phi(x)}{(\mu + \sigma x) \{1 - \Phi(x)\} } = \lim_{x \to ...
0
votes
1answer
25 views

Raising Polar to the Power

Ok guys, I'm back! This is gonna be my last question here, as I have practically finished test review. Raise to the the power: b ) $(8 cis(120 ^\circ)))^{1/3}$ ... Well for this there are two ...
-2
votes
1answer
45 views

Why does it not have another intersecting point in this graph [closed]

PROBLEM : You are about to take a test that contains questions of type A worth $4$ points and of type B worth $7$ points. You must answer at least $5$ of type A and $3$ of type B, but time restricts ...
4
votes
4answers
100 views

Find the two square roots of $i$

I have this question I am stumped upon for my Test-Review: Write $i$ as a complex number in polar form. Use the result and DeMoivre's Theorem to find the square roots of $i$. I got the first ...
-2
votes
3answers
54 views

Converting into rectangular form

I have 2 related questions: First: Let $z_1 = 2+2i$ and $z_2 = 2-2i$. Find $z_1z_2 $ in rectangular form. I have no idea... I'm also clueless about this question: Change the following to ...
-1
votes
2answers
65 views

Simplify: $S=3^{1/3}\cdot 7^{1/4}$

Simplify: $$S=3^{1/3}\cdot7^{1/4}$$ How is it possible to simplify this? The exponents are completely different.
-4
votes
4answers
45 views

Let $a=b+2$. suppose the sum of $a$ and $b$ is equal to the product of $a$ and $b$. Prove that $a$ and $b $are not integers [closed]

Question. Let $a=b+2$. suppose the sum of $a$ and $b$ is equal to the product of $a$ and $b$. Prove that $a$ and $b$ are not integers. Any help is very much appreciated!
3
votes
6answers
62 views

Simplify $\frac{1}{\sqrt{4+2\sqrt{3}} - \sqrt{4-2\sqrt{3}}}$

Simplify $$\frac{1}{\sqrt{4+2\sqrt{3}} - \sqrt{4-2\sqrt{3}}}$$ I know there is another easier method except the one I answered. I cannot find it. Can you please help? Thanks in advance.
0
votes
2answers
29 views

Mathematical syntax when defining domain and range

What is the correct syntax when writing about the domain and range of a function? For example, let's say: $f(y)=\cos{y}$ and $y(x)=\arcsin{x}$ In order to simplify the function ...
3
votes
0answers
56 views

Find the polynomial $p(x)$

A polynomial $p(x)$ gives a remainder of $1$ when divided by $x^{100}$ and a remainder of $2$ when divided by $(x-2)^3$. Evaluate $p(x)$. By the Remainder Theorem, $p(x)$ can be written as ...
4
votes
1answer
56 views

How to solve $f(x)+f\left(\frac{1}{x}\right)=e^{x+\frac{1}{x}}$

Given that $f:\mathbb{R}_0 \rightarrow \mathbb{R}_0$ find such $f$ that $$f(x)+f\left(\frac{1}{x}\right)=e^{x+\frac{1}{x}}$$ Note that I came up with this question, and personally am not sure ...
0
votes
4answers
90 views

Exponent of a number is a square root?

Say you have a number $x^{\sqrt 2} = 2$. Is there any way to find $x$ without resorting to approximation (i.e. using some sort of formula instead)? EDIT: The question seemed a bit vague, and frankly ...
-1
votes
0answers
32 views

A dog costs $\$d$ per month to feed, a cat costs $\$27$ less per month to feed. What is the cost of feeding two cats and a dog for a month?

A dog costs $\$d$ per month to feed, a cat costs $\$27$ less per month to feed. What is the cost of feeding 2 cats and a dog for the month. I'm trying to help my son understand this algebra ...
5
votes
2answers
75 views

Radical equation $\sqrt{x+1}+\sqrt{x-1}-\sqrt{x^2 -1}=x$

The Question: $\sqrt{x+1}+\sqrt{x-1}-\sqrt{x^2 -1}=x$ Only thing I can take from this is that $x^2 -1=(x+1)(x-1)$, but I don't think that would help in any way. I know the answer, but I don't know ...
0
votes
0answers
15 views

Graph method for representing multiplicity of solutions

Inspired by Degenerate zeroes Fundamental theorem of Algebra In Cartesian coordinates if a root has a multiplicity of even then it "bounces" back up from the $x$-axis, if the root has a multiplicity ...
1
vote
1answer
29 views

Degenerate zeroes, fundamental theorem of algebra.

The theorem is also stated as follows: every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n roots. The equivalence of the two ...
3
votes
2answers
35 views

Adding Polynomials with exponents. Can't get same answer as answer key.

This question is from my final practice exam. simplify and express each answer using positive exponents only. $$(2x^3y^{-2}z^0)^2+8x^{-3}y^2 $$ After working out the problem this is the answer that ...
1
vote
2answers
40 views

If $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$

I am trying to find a simple way to show the fact that if $a\le b\le c$, then $|b-d|\le\max\{|a-d|,|c-d|\}$ for any number $d$. Is there a way to do this besides breaking it up into the cases 1) ...
0
votes
1answer
27 views

Total Percentage Change - Why does this work?

Algebraically this works, but I'm looking to understand (1) why it works and (2) if there is a simpler formula. Problem: I want to get total % change over time. Facts: Example Data - Jan 2013 - ...
1
vote
2answers
21 views

Annual Inflation Rate Problem

This question has me stumped. Anyone have ideas to point me in the right direction of a quick and easy way to solve this? I took this from a practice GRE subject math test and each problem is only ...
3
votes
6answers
91 views

How would you tackle $\sin(2x)-2\sin(3x)+\sin(4x)=0$?

Just wondering how would you tackle this trig equation. I tried a lot of different approaches but I end up expanding. Even my prof couldn't answer it earlier. I'm just curious . $$ ...