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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5
votes
1answer
94 views

Factoring $a^2+b^2$?

I remember there was a way to factor $a^2+b^2$ into something along the lines of $(a+\sqrt{a}+b)(a-\sqrt{a}+b)$ . I tried every combination of pluses and minuses for this form, but I couldn't get back ...
0
votes
4answers
56 views

If ${\overline{z}}^2=z^2$ where z is a complex number then z is either real or pure imaginary

If ${\overline{z}}^2=z^2$ where z is a complex number then z is either real or pure imaginary Approach: I approach this algebraically. I set $z=x+yi$ and came up with $(x^2-y^2)-2xyi=(x-y^2)+2xyi$ I ...
-1
votes
0answers
5 views

rectilinear motion and the derivative as a rate of change

If A(x) square centimeters is the area of a square having a side of x centimeters, use a calculator to compute the average rate of change of A(x) with respect to x as x changes from (a) 4.000 to 4.600;...
0
votes
1answer
32 views

Proof about triangle inequality

Follow the steps bellow to give an algebraic derivation of the triangle inequality $$|z_1+z_2| \leq |z_1|+|z_2|$$ a) Show that $$|z_1+z_2|^2=(z_1+z_2)(\overline{z_1}+\overline{z_2})=z_1\overline{z_1}...
2
votes
1answer
57 views

Name this 2-dim Parametric Curve

is there a name for the following parametric curve? Thanks. I encountered it when playing around with the tangent lines and normal lines of an ellipse. The parameter $\theta$ is the same parametric ...
0
votes
3answers
28 views

Solving a difficult logarithmic inequality involving a fraction

My question regards solving the following logarithmic inequality for x: $$\dfrac{\log_{2} (x^{2}-6x+8)}{\log_{2} (x-8)}< 1.$$ I have become very confused as to how to solve more complicated ...
2
votes
7answers
123 views

Why when $x^2=y^2$ then $x=y$ doesn't hold sometimes but $x^3=y^3$ then $x=y$

Why when $x^2=y^2$ then $x=y$ doesn't hold sometimes , but $x^3=y^3$ then $x=y$ holds in the real numbers. . I don't understand a thing If we do like this: $\sqrt{x^2}=\sqrt{y^2}$then $x=y$ but why ...
2
votes
1answer
21 views

Sketch the set of points determined by the following condition

$|2 \overline{z}+i|=4$ if I let $z=x+yi$, I got $4x^2+(1-2y)^2=16$ I don't know if that's right. Should I just plot that in wolfram alpha?
3
votes
2answers
67 views

Why doesn't transposing matter?

I was helping a friend with linear algebra, particularly I was teaching how to check if a collection of vectors $v_1, ..., v_k \in \mathbb{R}^n$ are linearly independent (assuming the vectors are ...
-5
votes
0answers
32 views

Find domain and range [on hold]

$F(x)= -\sqrt{x+3}$ I do not know how to find domain and range so I have tried nothing. Please help me and explain how you find domain and range.
0
votes
3answers
84 views

Solve for $x$ :- $(\sqrt{3x^2+6x+7} + \sqrt{5x^2+10x+4}) = 4-2x-x^2$

The main question is :- Solve for $x$ :- $$(\sqrt{3x^2+6x+7} + \sqrt{5x^2+10x+4}) = 4-2x-x^2$$ My approach :- For convenience, I assume $$t_1=3x^2+6x+7$$ $$t_2=5x^2+10x+4$$ Then, by rationalizing,...
-5
votes
3answers
38 views

Inequality troubles $\frac{x^2(5+x)(x-4)}{(x+2)(x-2)}\geq 0$ [on hold]

What $x$ satisfy the inequality: $$\frac{x^2(5+x)(x-4)}{(x+2)(x-2)}\geq 0$$ I have never been good at math, more of a right brained person. Can anybody help me to solve this inequality?I have been ...
-1
votes
0answers
13 views

General non-numerical solution methods for arbitrarily given non-recursive transcendental closed-form equation of one variable?

Which general non-numerical methods are there to solve an arbitrarily given non-recursive real or complex transcendental closed-form equation of one variable? An example of such an equation is: $$3 - ...
0
votes
1answer
35 views

Making $t$ a subject of equation?

In this equation I want to make the variable $t$ a subject. Is this possible? $$s = ut + \frac12 at^2$$
29
votes
11answers
970 views

What are the Laws of Rational Exponents?

On Math SE, I've seen several questions which relate to the following. By abusing the laws of exponents for rational exponents, one can come up with any number of apparent paradoxes, in which a number ...
-1
votes
1answer
84 views

Tricky question involving binomial expansion [on hold]

For given $m$, what is the highest power of $2$ that divides $[(\sqrt3 +1)^m]+1$? where $[x]$ denotes the greatest integer less than or equal to $x$. I have no clue how to proceed.
0
votes
2answers
28 views

Help me learn how effective this new medication is:

I can't quite understand exactly how effective this med is because I don't know the proper equation. Here is what I know: A medication called Rebif has proven to be 67% better than a placebo. This NEW ...
1
vote
1answer
22 views

A certain colony of bacteria

A certain colony of bacteria is growing at 4% daily. If the culture of bacteria has a population of 4,000 today, what will it be after 10 days?
3
votes
2answers
37 views

Algebraic Expressions with Fractions

can someone review this and see if i've done it correctly please. $$\frac{\frac {3x} {y}}{\frac {2x}{7}} $$ $$= \frac{3x}{y} . \frac{7}{2x}$$ $$= \frac{21x}{2xy} $$ $$= \frac {21}{2y} $$ Thank ...
3
votes
1answer
51 views

How to describe a summation of $\frac{1}{2^x3^y}$ and evaluate.

I want too calculate the value of this sum: $$\sum \frac{1}{2^x3^y}$$ Where we sum up all permutations of terms involving a nonnegative integer $x$ and a nonnegative integer $y$. How can I ...
0
votes
3answers
26 views

Mathematical Induction and Algebra

In a question taken from Discrete Mathematic With Applications A question tries to prove $2^{2n}-1$ is divisible by 3. In the solution it has $$2^{2k}(3+1)-1$$ $$2^{2k}.3+(2^{2k}-1)$$ What ...
4
votes
1answer
112 views

Algebraic solution for the value of $x$.

I solved this problem the fifteen years ago without numerically solving equations of degree 4, I was happy in a substitution that I avoid directly attacking equations of degree 4. Today my nephew, ...
1
vote
2answers
70 views

How to express the rest of division by three, with very elementary functions?

Is it possible to express $\; "\!n\pmod 3\!"\;$ with combinations of the functions plus, minus, multiplication, division and exponentiation in $\mathbb C$ or preferably in $\mathbb Z[i]$? I'ts not ...
7
votes
3answers
268 views

Partial fractions and using values not in domain

I'm studying partial fraction decomposition of rational expression. In this video the guy decompose this rational expression: $$ \frac{3x-8}{x^2-4x-5}$$ this becomes: $$\frac{3x-8}{(x-5)(x+1)} = \...
1
vote
1answer
39 views

properties of ratio and rule of cross multiplication

If $$\frac{l}{\sqrt a-\sqrt b}+\frac{m}{\sqrt b-\sqrt c}+\frac{n}{\sqrt c-\sqrt a} =0$$ $$\frac{l}{\sqrt a+\sqrt b}+\frac{m}{\sqrt b+\sqrt c}+\frac{n}{\sqrt c+\sqrt a} =0$$ Show that $$ \frac{l}{(a-b)(...
8
votes
5answers
125 views

Proving that the roots of $1/(x + a_1) + 1/(x+a_2) + … + 1/(x+a_n) = 1/x$ are all real

Prove that the roots of the equation: $$\frac1{x + a_1} + \frac1{x+a_2} + \cdots + \frac1{x+a_n} = \frac1x$$ are all real, where $a_1, a_2, \ldots, a_n$ are all negative real numbers.
0
votes
2answers
101 views

The English mathematician Augustus DeMorgan, who lived in the 19th century, once remarked that he was $x$ years old in the year x^2. When was he born?

I found this from Elementary Number theory from Koshy. The answer is $1806$, but does know tell how to find? Anyone have any idea how to solve this problem ?
0
votes
2answers
44 views

Trigo equation $3\cos^2(2x)=1+\sin x$?

How to solve $3\cos^2(2x)=1+\sin (x)$? $0\leq x \leq 360$ I've been figuring this question for a such time. My lecturer told me this was a wrong question, I didn't know why.
3
votes
4answers
84 views

Solving Quadratic system of equations

Solve this system of equations: $$(1) \quad 0=-10x^2-9xy+50x-25y-7y^2+5$$ $$(2) \quad 0=-5x^2-17xy+25x+50y-14y^2+7$$ Shame on me but I'm failing to solve this system. I can't see a short (...
0
votes
0answers
19 views

Discrepancy between range of amplitude of complex number and range of $\arctan$ function

I have a conceptual doubt regarding the amplitude of complex numbers. My teacher said that in general for a complex number $x+iy$ the amplitude can be given by $\tan^{-1}({\frac{y}{x}})$ in case we ...
1
vote
9answers
350 views

How to factor $9x^2-80x-9$? [on hold]

How do I factor a trinomial like this? I'm having a lot of difficulty. How do I deal with the $9x^2$?
1
vote
2answers
30 views

Question about surjective functions

Let $S$ and $T$ be sets and let $f : S \to T$ be a function. Prove the following: If $R$ is a set and $h:R \rightarrow S$ is a function such that $f \circ h$ is surjetive then also $f$ is ...
0
votes
0answers
36 views

Question about identities and bijective functions

Let $S$ and $T$ be sets and let $f : S \to T$ be a function. Prove the following: If $g,h:T \rightarrow S$ are functions satisfying $g \circ f =Id_S$ and $f \circ h=Id_T$ then $f$ is bijective ...
0
votes
1answer
26 views

Question about injective functions

Let $S$ and $T$ be sets and let $f : S \to T$ be a function. Prove the following: If $U$ is a set $g:T\rightarrow U$ is a function such that $g \circ f$ is injective then also $f$ is injective ...
2
votes
3answers
50 views

How does one plug radicals with non-perfect squares and variables into the Pythagorean theorem formula?

I am working on the following integral $$\int\left( 7x^2 - 3 \right)^{\frac 5 2} \, dx$$ I want to use the $\sqrt{u^2 - a^2}$ $u = a\sec\theta$ I know in order to get it into the form that will ...
3
votes
5answers
58 views

Implied plus-minus sign in radical equation?

Say we have: $$\sqrt{x+7}=5-x$$ Is it implicitly understood that the following also holds? $$-\sqrt{x+7}=5-x$$ I'm exploring the notion of "extraneous solutions." In this example, solving either ...
2
votes
2answers
80 views

How do I solve equation $\bar{z} = |z|$ correctly?

I'm having troubles, finding how solution would look like for complex equation of the form $\bar{z} = |z|$. Taking $z = x + iy$, we get the following: $$x - iy = \sqrt{x^2 + y^2},$$ then raising it to ...
0
votes
3answers
19 views

Find Rate when compound interest for successive years are given. [on hold]

A certain sum is lent a CI. The interest earned in 2 years is 272. The interest earned in 3 years is 439. Find rate of interest? Please tell me shortcut(if possible) of these type of questions for ...
0
votes
1answer
126 views

Find integer solutions

Find all integer solutions to the following: $2x+10y-11z=1$ $x-6y+14z=2$ I am not quite sure how to do this... I know I will get equations in the end with each variable expressed in terms of ...
0
votes
4answers
52 views

For what $k$ is $f(x) = kx^2-2x+k$ negative for all values of $x$?

What are the values of $k$ for which the quadratic function $f(x) = kx^2-2x+k$ is negative for all values of $x$? The values of $k$ should definitely be negative.
3
votes
1answer
53 views

Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$ [duplicate]

Simplify the expression $\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots +\binom{n+k}{k}$ My attempt: Using the formula $\binom{n+1}{k}=\binom{n}{k}+\binom{n}{k-1}$ $\binom{n}{0}+\binom{n+1}{1}+\...
3
votes
6answers
118 views

Evaluate the expression $\sqrt{6-2\sqrt5} + \sqrt{6+2\sqrt5}$

$$\sqrt{6-2\sqrt5} + \sqrt{6+2\sqrt5}$$ Can anyone tell me the formula to this expression. I tried to solve in by adding the two expression together and get $\sqrt{12}$ but as I insert each ...
35
votes
17answers
8k views

Prove $0! = 1$ from first principles

How can I prove from first principles that $0!$ is equal to $1$?
0
votes
1answer
26 views

Conversion of Excel formula back to written Formula…if possible!

I turn to you as a last resort, in the hope that you will forgive my Maths ignorance and help me with what may be a simple problem. Essentially, this involves converting an excel formula to written ...
1
vote
0answers
34 views

Are constants a special case of coefficients?

What I hope to understand better, is the relation between constants and coefficients. Consider the following polynomial: $$3x^2+2x+5$$ What are the coefficients in the expression? Obviously, 3 and 2 ...
0
votes
2answers
30 views

On the maximal of polynomial at a point

I faced this problem when I studied polynomial. Let $p(x)=ax^3+bx^2+cx+d$ be a cubic polynomial with real coefficients, and $p(5)+p(25)=1906$. Find the maximal value of $|p(15)|$. I ...
0
votes
2answers
28 views

Verify that $\sqrt{2}|z| \geq | R_z|+|Im_z|$

Verify that $\sqrt{2}|z| \geq | R_z|+|Im_z|$, suggestion: Reduce this inequality to $(|x|-|y|)^2 \geq0$ (z is a complex number. R stands for real part and Im stands for imaginary part) Approach: Let $...
0
votes
1answer
46 views

Prove that if $z_1*z_2$=0 then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers

Prove that if $z_1*z_2=0$ then at least one of $z_1$ and $z_2$ must be 0. $z_1$ and $z_2$ are complex numbers by using the following property: $|z_1z_2|=|z_1||z_2|$ Approach: if $z_1*z_2=0$ then $$|...
0
votes
4answers
60 views

Decomposition of $2x^2+9x-5$

So obviously one way is to find the roots of the polynomial which is $$\frac{-9\pm\sqrt{9^2-4 \cdot 2 \cdot (-5)}}{2 \cdot 2}=\frac{-9\pm 11}{4}$$ So we get $x_{1}=\frac{1}{2}$ and $x_{2}=-5$ so ...
0
votes
2answers
23 views

Properties of exponents when dealing with induction.

This will most likely be a simple question for most of you. While watching my lecture today the white board cut out and the instructor didn't explain the final step in an example. He went from $(3^{2^...