# Tagged Questions

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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### 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 ...
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### 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 ...
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### 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;...
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### Making $t$ a subject of equation?

In this equation I want to make the variable $t$ a subject. Is this possible? $$s = ut + \frac12 at^2$$
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### 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 ...
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### 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.
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### 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 ...
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### 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?
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 ...
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### 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.
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### 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 ?
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### 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.
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### 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 (...
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### 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 ...
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### 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 $9x^2$?
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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 ...
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### 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 ...
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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 ...
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### 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 ...
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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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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