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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-2
votes
2answers
36 views

Proving the equation has no root.

How to show that for $a\in \mathbb R$, the equation $x^2+12a^2+4ax-8a+8=0$ has no root?
4
votes
2answers
49 views

$3^x + 4^y = 5^z$

This is an advanced high-school problem. Find all natural $x,y$, and $z$ such that $3^x + 4^y = 5^z$. The only obvious solution I can see is $x=y=z=2$. Are there any other solutions?
2
votes
4answers
40 views

Sum of Series as $1,(2),1,(2,2),1,(2,2,2),1,(2,2,2,2),1…$

The Sum of First $2015$ terms of the Series... $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,.......................$ $\bf{My\; Try::}$ We Can Write the Given Series as ...
5
votes
3answers
47 views

“Rationalizing the denominator” of $1/(a + b\sqrt[3]{2} + c\sqrt[3]{4})$?

If $(a, b, c) \in \mathbb{Q}^3 \setminus \{(0, 0, 0)\}$, so that $a + b\sqrt[3]{2} + c\sqrt[3]{4}$ is a nonzero element of $\mathbb{Q}(\sqrt[3]{2})$, is there a formula for $${1\over{a + b\sqrt[3]{2} ...
-3
votes
0answers
29 views

Change the subject of a formula [on hold]

$150 \cdot 10^6 = \dfrac{3pR^2}{4t^2}$ How do I find out what $t$ is, hence make it the subject of the equation. I think I know what the answer should be: $p=1.5 \cdot 10^6$ $R= 0.075$ ...
1
vote
1answer
18 views

Question in regard to solving for inverse laplace transform

I am having some confusion when it comes to solving for the inverse laplace transform. ( We are allowed the tables with the common values by the way). Il give an example. Take, ...
-1
votes
0answers
19 views

Excel Exponential Line Values [on hold]

I am trying to use Excel to graph an Exponential Trendline, but would also like to extract the values to be used in a spreadsheet I am developing. The trendline formula is showing as: $y = ...
-4
votes
2answers
37 views

Logarithm with nth root [on hold]

I made it but the result is very strange. I want every step to the result $$ \large 6\log_{10}\frac{\sqrt2}{\sqrt[3]{3+\sqrt5}} $$
1
vote
1answer
25 views

Expanding a term with a sum

We have the following quantity: $$E\left[\left(\sum^n_{j=1} (X(t_j) - X(t_{j-1}))^2-t\right)^2\right]$$ My textbook says this can be expanded in the following way (colors are my touch) ...
-4
votes
0answers
23 views

solve the equation for the maximum positive integral value [on hold]

$$\large\displaystyle \sum_{x=1}^{\infty} \displaystyle \log_{n}\left(\frac{(x+a-1)(x+a+1)}{(x+a-3)(x+a+3)}\right)=1$$ How do I solve the above equation for the maximum positive integral value of $n$ ...
1
vote
0answers
23 views

Simplifying cyclometric function

How does one simplify this function? $$ f(x) = \arccos(\frac{\pi}{2} - \sin(x)) $$ A plot in GeoGebra showed a graph that looked like semicircle, so can one expect something in this form: ...
-4
votes
2answers
54 views

Simplifying the expression $\frac{x + y}{x - y} + \frac{1}{x + y} - \frac{x^2 + y^2}{y^2 - x^2}$

Can you tell me why my answer is wrong? $$\frac {x+y} {x-y} + \frac 1 {x+y} - \frac {x^2+y^2} {y^2-x^2} = \frac {x^2 + y^2} {x^2-y^2} + \frac {x-y} {x^2-y^2} + \frac {x^2+y^2} {x^2-y^2} = 2x^2 + 2y^2 ...
-8
votes
1answer
47 views

How many? I need help, please help me. [on hold]

I need help with this right now. How many $\$$ is $100\%$, if $25\%$ is $15\$$.
-4
votes
1answer
49 views

How to simplify $ \frac{x^2-9x+14}{x^2+7x+12} \div \frac{3x^2-21x}{4x^3+16x^2} $? [on hold]

I'm having trouble simplifying a fraction: $$ \frac{x^2-9x+14}{x^2+7x+12} \div \frac{3x^2-21x}{4x^3+16x^2} $$ I tried it but I think my factoring is wrong keep coming out wrong answers.
0
votes
3answers
37 views

How do I factorise the following expression?

How do I go from the left expression to the right one? $$ (2-x)^2 \cdot (-2-x) - (-2-x) = - (x+2)(x-3)(x-1) $$ I'm guessing that I have to solve the third degree equation. What are the steps for ...
0
votes
0answers
16 views

Simplifying $\theta u^2 + (1-\theta)v^2 - [\theta u + (1-\theta)v]^2$

I've been working through a problem in Chiang's Fundamental Methods of Mathematical Economics and I ran into a little bit of trouble. So the problem is to check whether a function is concave or convex ...
-1
votes
1answer
75 views

Solve this equation for x [on hold]

I've come up with an equation whilst solving a problem but I need to rearrange it for $x$. Putting it in Wolfram Alpha doesn't give me anything. This is the equation $$(1+x)^c - (1-x)^c = d.$$ $c,d$ ...
-1
votes
0answers
41 views

Interesting and challenging problem [on hold]

I've been given this problem to solve, but didn't succeed until now. Can you help me? A city has 5 billion paper money (bills) in circulation. Thirty million of them are taken daily to the bank ...
0
votes
1answer
26 views

Summation operation for precalculus

Studying Spivak's Calculus I came across a relation I find hard to grasp. In particular, I want to understand it without using proofs by induction. So please prove or explain the following ...
-4
votes
1answer
25 views

Find the domain and range of $f$ and $f^{-1}$ [on hold]

Find the domain and range of $f$ and $f^{−1}$ $f(x) = x^2 − 9, \ \ \ x \le 0$ $f^{−1}(x) = -\sqrt{x+9}$
-7
votes
1answer
53 views

Simplify $2^3-3^{\frac{5}{8}}+2^2+3^{\frac{5}{8}}+2^1$ [on hold]

How can I simplify this expression? I really need to know how. $2^3-3^{\frac{5}{8}}+2^2+3^{\frac{5}{8}}+2^1$
-1
votes
1answer
41 views

Is this Factored out fully? (Exponents) [on hold]

$2x^2 + 32$ $\Rightarrow$ $2(x + 4)^2$ Is this correct?
4
votes
7answers
82 views

Calculate $\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$

Question: Calculate $$\lim_{x \to 0} \frac{e^{3x} - 1}{e^{4x} - 1}$$ using substitution, cancellation, factoring etc. and common standard limits (i.e. not by L'Hôpital's rule). Attempted ...
0
votes
0answers
17 views

transforming an equation into a difference equation

I know how to rewrite a differential equation into a difference equation using Euler's forward difference. However, I'm at a loss as how to convert a given equation into a difference equation. For ...
3
votes
3answers
274 views

Solving a Radical Equation $5(\sqrt{1-x} + \sqrt{1+x}) = 6x + 8\sqrt{1-x^2}$ (squaring doesn't help)

How should I approach this problem: $$ 5(\sqrt{1-x} + \sqrt{1+x}) = 6x + 8\sqrt{1-x^2} $$ I've tried squaring both sides but to get rid of all the radicals requires turning it into a quartic equation, ...
2
votes
2answers
18 views

How do I use the $A= Pe^{rt}$ formula in this question?

So the question is in $2000$ the deer population in a certain area was $800$. The number of deer increases exponentially at a rate of $7%$ per year. Predict the population in $2009$. a) $1408$ b) ...
0
votes
1answer
34 views

Working out “break even” point

Please bear with me, my brain is hopeless at math. My colleague has a Jeep Grand Cherokee. He had a chip installed which cost him $\textrm{ZAR}3500$. He wants to know when his purchase of the chip ...
0
votes
1answer
43 views

Confusion with Summations

I am having a little bit of confusion regarding summations. I know that $$\sum_{i=m}^n a_i = a_{m}+a_{m+1}+\cdots +a_{n-1}+a_n$$ Here is my confusion. How do we interpret/decompose the following: ...
-3
votes
0answers
30 views

Synthetic division [on hold]

This is about changing fractions into a mixed expression. So I have to do divide them. But I don't know why this problem tells me to leave spaces! Here is the problem: $\dfrac{k^3 - 1}{k - 1}$ ...
3
votes
4answers
358 views

How to solve certain types of integrals

I'm asking for a walk through of integrals in the form: $$\int \frac{a(x)}{b(x)}\,dx$$ where both $a(x)$ and $b(x)$ are polynomials in their lowest terms. For instance $$\int ...
0
votes
1answer
88 views

Why is $\sqrt{X}\times\sqrt{X}=X$?

Today I was solving the limit $(\ln(x))/(2*(x^{1/2})$ but then faced the step after applying the derivation that ended up with $(1/x)/(1/x^{1/2})$ and the result of that was $1/x^{1/2}$. When I asked ...
0
votes
2answers
41 views

Proof of $xyz+1= 2yz$, Given $x=\log_{2a}a$, $y=\log_{3a} 2a$, $z=\log_{4a} 3a$ [on hold]

Proof of $xyz+1= 2yz$, Given $x=\log_{2a}a$, $y=\log_{3a} 2a$, $z=\log_{4a} 3a$
1
vote
3answers
19 views

Complex plane (Show that triangle is right-angled)

The points $O$,$P$ and $Q$ in the complex plane represent the complex numbers $0+0i$, $4+2i$ and $3-i$ respectively. Find the exact length of $PQ$ and hence, or otherwise, show that triangle $OPQ$ is ...
1
vote
1answer
32 views

Complex Number (Angle)

The complex number $z$ is given by $z=-2+2i$ Find the modulus and argument of $z$ Write down the modulus and argument of $\frac{1}{z}$ Show on an Argand diagram the points A,B and C representing the ...
3
votes
1answer
30 views

Proof by induction from Spivak's calculus ch 2- 3b

I was cracking my head over the following proof (by induction) from Spivak's calculus. Givens: $ \binom{n+1}{k}=\binom{n}{k-1}+\binom{n}{k} $ and $ n \ge k $ Task: Proof by induction that $ ...
0
votes
1answer
29 views

Complex Numbers (Find p and q)

The complex numbers z1 and z2 are given by $$z_1=5+i,z_2=2-3i$$ Determine the values of the real constants $p$ and $q$ such that $$\frac{p+iq+3z_1}{p-iq+3z_2}=2i$$ My attempt, I substitute $z_1$ and ...
0
votes
3answers
110 views

Complex Numbers (Modulus)

The complex numbers $z_1$ and $z_2$ are given by $$z_1=5+i,z_2=2-3i$$ Find the modulus of $z_1-z_2$ My attempt, modulus of $z_1-z_2=\sqrt{5^2+1^2}-\sqrt{2^2+3^2}$ $=\sqrt{26}-\sqrt{13}$ ...
0
votes
1answer
33 views

vertices of a hyperbola the silliest question ever

I'm given that the center of the hyperbola is $(2,1)$ and $a=3$ and asked to find the vertices. Since vertices are on the same line with the axis of symmetry I thought the coordinates should be $(2,1 ...
2
votes
1answer
58 views

Solutions of $\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$

$\sqrt{x+4+2\sqrt{x+3}}-(x^2+4x+3)^{1/3}=1$ I get that $-3$ as a solution, but apparently 1 is as well a solution, and I don't see a mechanism through which I could find it. Any help would be ...
0
votes
3answers
62 views

Prove that A is invertible if $A^2 - 4A -7I = 0$. [duplicate]

The $2 \times 2$ matrix $A$ satisfies $$A^2 - 4A -7I = 0,$$ where $I$ is the identity matrix. Prove that $A$ is invertible. I'm not sure how to do this. Help would be appreciated.
0
votes
1answer
20 views

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there… [duplicate]

Let ${A}$ be a $2 \times 2$ matrix. For every two-dimensional vector ${v}$, there exists a two-dimensional vector ${w}$ such that Aw = v. Show that ${A}$ is invertible. I'm not sure how to do this.
1
vote
1answer
96 views

Proving that matrix in equation is invertible

The $2 \times 2$ matrix ${A}$ satisfies ${A}^2 - 4 {A} - 7 {I} = {0}$ where ${I}$ is the $2 \times 2$ identity matrix. Prove that ${A}$ is invertible. I have tried to solve it like a quadratic, but ...
0
votes
2answers
30 views

Solving $f(x) = x^5 +x + 1 = 0$ with halving the interval / bisection method

Question: Use halving the interval / bisection method to approximately solve: $$f(x) = x^5+ x + 1 = 0$$ with a precision of $\pm 0.1$ Attempted solution: The general idea, as I understand it, is ...
0
votes
2answers
18 views

Complex number (Rhombus)

Given that $z_1=1+2i$ and $z_2=\frac{3}{5}+\frac{4}{5}i$, write $z_1z_2$ and $\frac{z_1}{z_2}$ in the form $p+iq$, where $p$ and $q \in R$. In an Argand diagram, the origin O and the points ...
-1
votes
1answer
22 views

Argument of Complex Number (Am I wrong?)

I'm given $z=-2+\sqrt{3}i$. So I worked out the argument of $arg(z)=\tan^{-1}(\frac{\sqrt{3}}{-2})$. I got the answer $2.256$rad. But the given answer is $2.45$rad. Am I wrong?
-1
votes
1answer
29 views

Doubt with Intervals and Inequalities

This doubt has been bothering me for ages. I would be truly grateful for any help. Problem 1: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals Solution: $x\in(2,4)\cup(4,6)$ ...
2
votes
3answers
29 views

Absolute Value Inequality Problem

Problem: $\dfrac{2}{|x-4|}>1$ Express the solutions using intervals My attempt using the Definition of Modulus: $$\dfrac{2-|x-4|}{|x-4|}>0$$ $$CASE A:x-4\ge 0\Rightarrow x\ge4\Rightarrow ...
1
vote
1answer
34 views

Simplify the expression and find the minimum value

I want to simplify the expression \begin{equation} A = \frac{\sqrt{1-\sqrt{1-x^2}}\Big[\sqrt{(1+x)^3} + \sqrt{(1-x)^3} \Big]}{2-\sqrt{1-\sqrt{1-x^2}}} \end{equation} and find the minimum value of ...
1
vote
1answer
28 views

On finite sums and products

I'd like to get a good book on finite summations and products before I study infinite series more in depth next year. The book should cover geometric/ harmonic sums and prove different formulas for ...
1
vote
0answers
28 views

Calculating a formula for variables with multiple values equaling the same total

I'm having a bit of trouble puzzling a formula for some code I'm using to develop a piece of software. I'm not very savvy with what the technical terms for all of what I'm describing are, but I'll try ...