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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3
votes
1answer
20 views

How to find the numbers that the product of it's digits is equal to ten times the sum of them.

I formulated this so that the number be in range $[111,999]$, it was narrowed so that $a,b,c$ is not equal to zero. $$a\cdot b\cdot c = (a+b+c)\cdot 10$$ With this we can see that $\frac{a\cdot ...
-1
votes
1answer
39 views

An algebra word problem in one variable on percentages - assistance required

I'm new here. I'm not sure how this works, but I was wondering if someone could help me with this problem. I'm having difficulty solving it, and I was hoping someone could help me. Thank you! Assign ...
0
votes
2answers
59 views

Word problem on the perimeter and sides of a triangle

Having difficulty answering this question. I'm not sure how to do it, but if anyone could show me the steps; so I could answer it myself, that would be great. Thanks Assign x, make an equation and ...
16
votes
3answers
104 views

Intriguing Indefinite Integral: $\int ( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1})^2 \mathrm{d}x$

Evaluate $$\int \left( \frac{x^2-3x+\frac{1}{3}}{x^3-x+1}\right)^2 \mathrm{d}x$$ I tried using partial fractions but the denominator doesn't factor out nicely. I also substituted ...
0
votes
0answers
9 views

Central orbits ellipse change of centre of force

a body is decribing an ellipse of eccenticity e under the action of central force directed towards the focus, and when at the nearer apse, the centre of force is transferred to the other focus, Find ...
2
votes
2answers
44 views

Advanced (for me) algebra and mean

I have been struggling over questions like these which my maths teacher has been throwing into our weekly papers for about a week now, and it has stumped all of us. Can you help? Question: There are n ...
-5
votes
1answer
43 views

A trick and interesting math SUM [duplicate]

If $x=(a/b)^{2ab/(a^2-b^2)}$ I want to prove that $$( (ab)/(a^2+b^2) )(x^{a/b} + x ^ {b/a})=(a/b)^{(a^2+b^2)/(a^2 - b^2)}.$$
3
votes
2answers
28 views

Stuck in a problem in permutation and combination.

I am solving problems in permutation & combination and stuck in this problem. Two players $P_1$ and $P_2$ play a series of $2n$ games. Each game can result in either a win or a loss for $P_1$. ...
-3
votes
0answers
56 views

Prove this enticing equation [on hold]

If $$x=\left(\frac{a}{b}\right)^{\frac{2ab}{a^2-b^2}}$$ I want to prove that $$\left(\frac{ab}{a^2+b^2}\right)\left(x^{\frac{a}{b}} + x ...
1
vote
1answer
23 views

Total no. of Transitive Relation on $A = \{a,b,c\}$

Calculation of total no. of Transitive Relation on $\displaystyle A = \left\{a,b,c\right\}.$ $\bf{My\; Try::}$ First We will calculate Total no. of Relation on $A$, Which is $\displaystyle = 2^{3^2} ...
7
votes
2answers
369 views

Is a circle classified as an ellipse?

I read that an ellipse had $2$ focal points. So, I thought if a circle had $2$ points that were simply infinitesimally close together wouldn't it be classified as an ellipse? Help would be ...
1
vote
2answers
33 views

Show that: $\sinh^{-1}(x) = \ln(x + \sqrt{x^2 +1 } )$

could someone Please give me some hint of how to do this question thanks
1
vote
3answers
45 views

Precalculus unit circle with imaginary axis.

(a) Suppose $p$ and $q$ are points on the unit circle such that the line through $p$ and $q$ intersects the real axis. Show that if $z$ is the point where this line intersects the real axis, then $z = ...
1
vote
1answer
51 views

Express $C_n = \cosh(0) + \cosh(1) + \cosh (2) + \dots + \cosh(n)$

Could someone give me some hint of how to do this question please. I've been stuck for more than $3$ hours on this question.
2
votes
3answers
70 views

Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$?

Could you explain the expansion of $(1+\frac{dx}{x})^{-2}$? Source: calculus made easy by S. Thompson. I have looked up the formula for binomial theorem with negative exponents but it is confusing. ...
-3
votes
1answer
20 views

Solving for a variable where the variable appears twice in the equation [on hold]

Introduction: Lean body mass (LBM) can be calculated by the formula $$x=z-(z\cdot y),$$ where $x=$LBM, $z=$total weight and $y=$percent body fat. Problem: Solve for $z$ if $x$ and $y$ are known. ...
0
votes
2answers
55 views

How to find minutes?

Need help solving this real life problem, I have an SD Card of $4$GB(gigabyte), and a $32$ second video occupies $6.12$MB(megabyte), I need to know how many minutes or seconds can this $4$GB SD Card ...
3
votes
2answers
35 views

Proving $1+5+9+\cdots+(4n+1) = (n+1)(2n+1)$ by induction (is there a typo?)

Using mathematical induction, prove that $$1+5+9+\cdots+(4n+1) = (n+1)(2n+1).$$ I understand the steps to take in order to prove by induction. It is also to my understanding that step 1 would be ...
4
votes
2answers
60 views

Proving that $3^n<n!$ when $n\geq 7$

It's been 10 years since my last math class so I'm very rusty. How would I go about proving $$3^n < n!$$ where $n \geq 7$? I understand that factorials grow faster than set values with a variable ...
0
votes
1answer
35 views

Basic function manipulation and simplification question for $f((x-f(x))^2)$

I've run into a bit of a wall trying to understand why the following two equations are equivalent: $$f((x-f(x))^2) = f(x^2)-f(x)^2$$ I'm running into this with calculating population variance in ...
1
vote
4answers
38 views

How does $\frac{-1}{x^2}+2x=0$ become $2x^3-1=0$?

Below is part of a solution to a critical points question. I'm just not sure how the equation on the left becomes the equation on the right. Could someone please show me the steps in-between? Thanks. ...
0
votes
2answers
118 views

Simplifying $\frac{\log(x)}{x}=y$.

I am trying to find the value of $r$ where the Rule of 72 will accurately estimate an investment's doubling time. Put simply, the Rule of 72 requires that 72 be divided by the interest percentage per ...
0
votes
2answers
68 views

Show that point you found for $x^2 - 2y^2 = 1$ using line through$ (1, 0)$ has integer coordinates.

The curve $x^2 - 2y^2 = 1$ includes the point $(1, 0)$. Let $L$ be the line through $(1, 0)$ with slope $m$. Find the other point where $L$ intersects the curve. Suppose that you take $m = v/u$, where ...
0
votes
3answers
53 views

I can't understand how to factor a quadratic

I can't figure out why some actions are possible in factoring $\frac{r–3}{r^2+11r–42}$. Factor the quadratic out of the denominator $\frac{r-3}{(r-3)(r+14)}$. This does not make sense how can $+11$ be ...
0
votes
0answers
12 views

moment generating function with Taylor series simplification

Denotes $a(0,r_1,r_2)$ as the annulus with radii $r_1<r_2$ centered at the origin $0$ Consider two bands $a(0,s,t)$ and $a(0,u,\sim)$ for $1\leq s\leq t\leq u$ Suppose a variable (call it an ...
2
votes
4answers
73 views

Rewrite $\sum_{i=0}^{n-1} (2i+1)=n^2$ to start induction from $k = 0$?

I'm trying to learn mathematical induction. The text asks for being totally rigorous i.e start induction from $k=0$. I want to prove that $$\sum_{i=0}^{n-1} (2i+1)=n^2,$$ i.e. the sum of the first ...
2
votes
1answer
44 views

Expansion of $(a+b+c+d)^n$

How to expand when $(a+b+c+d)^n$? Can I allow it to be $[(a+b)+(c+d)]^n$ and then use binomial theorem to expand it?
1
vote
1answer
47 views

Algebraic solution for a equation involving radicals?

$$f(x)=-\frac{5-x}{\sqrt{(5-x)^2+4}}-\frac{5-x}{\sqrt{(5-x)^2+9}}+1$$ I'm having trouble solving $f(x)=0$. I know there is a root for this function, I took the steps to set $f(x)$ to be equal to ...
0
votes
2answers
36 views

Solving for $x$ after simplifying, please check my work! [duplicate]

I was presented with the question: $$(\frac{x}{a})^{3.2} + (\frac{y}{b})^{3.2} = 1$$ While: $$\frac{a}{b} = \frac{174.1}{86}=c$$ I began to simplify and reached: $$x = c(b^{3.2}-y^{3.2})^{1/3.2}$$ ...
5
votes
2answers
59 views

Evaluate $\text{k}$ from the given equation

If $$ \int_{0}^{\infty} \left(\dfrac{\ln x}{1-x}\right)^{2} \mathrm{d}x + \text{k} \times \int_{0}^{1} \dfrac{\ln (1-x)}{x} \mathrm{d}x =0$$ then find the value of $\text{k}$ My ...
0
votes
1answer
34 views

How to solve this system of equations of degree 3?

I was solving a problem and after making some substitutions I got these equations- $$a^3-2a^2+a(b^2+1)+8b-2b^2=0$$and$$b^3+(a^2-1)b+8a=0$$ I can't solve these equations. If they can't be solved, then ...
0
votes
1answer
25 views

Question in finding final position in circle

im having difficulty solving this problem right now, I tried couple of times but i couldnt figure it out, can you please help and explain? thank you
4
votes
4answers
53 views

Find all integers n such that the quadratic $5x^2 + nx – 13$ can be expressed as the product of two linear factors with integer coefficients.

I am unsure of how to approach this problem. I have thought about using the Rational root theorem, but I am unsure if this answers the question being asked. Using the theorem, I get $\frac{p}{q} = ...
3
votes
2answers
40 views

Complicated Planar Geometry [duplicate]

A regular polygon $\mathcal{P}$ is inscribed in a circle $\Gamma$. Let $A$, $B$, and $C$ be three consecutive vertices of the polygon $\mathcal{P}$, and let $M$ be a point on the arc $AC$ of $\Gamma$ ...
0
votes
1answer
16 views

The Average Rate of Change of Function

Here is the question: The following chart shows the growth of a crowd at a rally over a 3 h period. Time (in hours): $ 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0 $ Number of people: $0 , 176, 245, 388, 402, ...
0
votes
2answers
54 views

Please check my simplification for x?

I'm hoping someone could check my work. I'm trying to simplify the equation for $x$. $$(\frac{x}{a})^{3.2} + (\frac{y}{b})^{3.2} = 1$$ $$\frac{a}{b} = \frac{174.1}{86}$$ Thus I began by substituting ...
0
votes
0answers
11 views

Need help with rearranging a first order condition

Given the following function, I have managed to derive the correct first order condition for optimality: (1) $\mathbb U(x)= \rho (w-x)^{\frac{1}{2}} + (1-\rho)(w+rx)^{\frac{1}{2}}$ (2) $ ...
2
votes
5answers
52 views

How to find point of intersection between quadratic and exponential equation

How can I find the point of intersection between a function like $2^x$ and $x^2$? I know you have to equate them but I don't know what to do after that.
0
votes
2answers
18 views

Sum of polynomial coefficient

Steve says to Jon, "I am thinking of a polynomial whose roots are all positive integers. The polynomial has the form $P(x)=2x^3-2ax^2+(a^2-81)x-c$ for some positive integers $a$ and $c$. Can you ...
2
votes
1answer
24 views

Show that if a circle with centre $(a,b)$ passes through the two points of intersection, then $a=0$.

For the equations $x^2+y^2=25$ and $x^2+(y-7)^2=18$ show that if a circle with centre $(a,b)$ passes through the two points of intersection, then $a=0$. I have sketched these equations in Wolfram ...
0
votes
1answer
23 views

scaling up data series based on polynomial equation

I have yearly time series data starting from 1990 to 2100 AD (x-axis). The value for 1990 is 0 and 2100 is 700, and it's increasing in each year (but not linearly). Based on this series, I come up ...
1
vote
2answers
82 views

Can we say that $\sqrt{2}=2/(2/(2/(2/\ldots)))$?

Can we say that $\sqrt{2}= \cfrac{2}{\cfrac{2}{\cfrac{2}{\cfrac{2}{\ldots}}}}$? We have ...
0
votes
0answers
22 views

What is the maximum value of $(q^{q+1} - 1)/({q^q}(q - 1))$ for integral $q \geq 5$?

What is the maximum value of $$\frac{q^{q+1} - 1}{{q^q}(q - 1)}$$ for integral $q \geq 5$? My attempt: The derivative of $$f(q) = \frac{q^{q+1} - 1}{{q^q}(q - 1)}$$ with respect to $q$ is $$f'(q) = ...
0
votes
1answer
12 views

Equivalent Interest Rates

Suppose that the APR on a certain product $x$ whose dollar value is $x_1$ is 5%. Now suppose we subtract some amount $x_2$ from $x_1$ where $x_2 < x_1$. Call this new amount $y$. How does one ...
0
votes
0answers
6 views

Circular Model- Finite Math [on hold]

Curtis-Roe Aviation Industries has two plants, I and II that produce the Zephyr jet engine used in their light commercial planes. The maximum capacity at Plant I is 100 units and at Plant II is 110 ...
-6
votes
1answer
151 views

How to prove or disprove this algebraic inequality?

How to prove or disprove the inequality (from math folklore) $$\sqrt{22(a^2+b^2+c^2)+5(ab+ac+bc)}\geq\sqrt{4a^2+ab+4b^2}+\sqrt{4b^2+bc+4c^2}+\sqrt{4c^2+ca+4a^2}$$ for nonnegative $a, b,$ and ...
3
votes
2answers
51 views

Basic Logs, simplifying

So basically the question goes: $\log_{14} 2 = a$, $\log_{14} 3 = b$, solve for $\log_7 24$. I have attached my work so far and the answer is ${3a+b}\over{1-a}$... I just got stuck at the end. Thanks
2
votes
1answer
46 views

Determining time it takes for two approaching cars to meet

I'm currently having problem with this question. "Solve the problem involving uniform motion: One car starts from the point A and moving to the point B. Another car starts at the same time from the ...
0
votes
1answer
20 views

Determine the value of y so that two line segments are parallel

Determine the value of $y$ so that the line segment with endpoints $P(3, y)$ and $Q(-3, -1)$ is parallel to the line segment with endpoints $R(-4, 9)$ and $S(5,6)$. I began by finding the slope ...
2
votes
0answers
28 views

Trigonometrical functions and complex numbers

(This question will at first appear too broad. However, the overall philosophy will be explained below in a way that asks specific questions, which I hope will be conducive to this being a reasonable ...