I am extremely new to mathematics, and I don’t have much training except for the basics so please excuse my rather basic question. The question itself: If I have the relationship $e^x + x – 2 = 0$; and $k$ is the number of solutions in $[0,1]$ and $n$ is the number of solutions not […]

I ran across this problem in a high school math competition: “You must use the integers $1$ to $9$ and only addition, subtraction, multiplication, division, and exponentiation to approximate the number $\pi$ as accurately as possible. Each integer must be used at most one time. Parenthesis are able to be used.” I tried writing a […]

There is a general formula for solving quadratic equations, namely the Quadratic Formula. For third degree equations of the form $ax^3+bx^2+cx+d=0$, there is a set of three equations: one for each root. Is there a general formula for solving equations of the form $ax^4+bx^3+cx^2+dx+e=0$ ? How about for higher degrees? If not, why not?

In grade school we learn to rationalize denominators of fractions when possible. We are taught that $\frac{\sqrt{2}}{2}$ is simpler than $\frac{1}{\sqrt{2}}$. An answer on this site says that “there is a bias against roots in the denominator of a fraction”. But such fractions are well-defined and I’m failing to see anything wrong with $\frac{1}{\sqrt{2}}$ – […]

One of the commonest mistakes made by students, appearing at every level of maths education up to about early undergraduate, is the so-called “Law of Universal Linearity”: $$ \frac{1}{a+b} \mathrel{\text{“=”}} \frac{1}{a} + \frac{1}{b} $$ $$ 2^{-3} \mathrel{\text{“=”}} -2^3 $$ $$ \sin (5x + 3y) \mathrel{\text{“=”}} \sin 5x + \sin 3y$$ and so on. Slightly more […]

I’m stuck at this algebra problem, it seems to me that’s what’s provided doesn’t even at all. Provided: $$a+b+c=0$$ Find the value of: $$\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$$ Like I’m not sure where to start, and the provided clue doesn’t even make sense. There’s no way I can think of to factor this big polynomial into a form like […]

I’m taking a course on Basic Conic Sections, and one of the ones we are discussing is of a parabola of the form $$y = a x^2 + b x + c$$ My teacher gave me the formula: $$x = -\frac{b}{2a}$$ as the $x$ coordinate of the vertex. I asked her why, and she told […]

I have a database of prices that already have a 10% discount. For example a product could be $100 after a 10% discount. Is there a reusable formula I can use to determine what the original price was of all the 10% discounted prices in the database? Edit: Thank you for the fast responses. Is […]

Prove that $x \neq 0,y \neq 0 \Rightarrow xy \neq 0$. Suppose $xy = 0$. Then $\frac{xy}{xy} = 1$. Can we say that $\frac{xy}{xy} = 0$ and hence $1 = 0$ which is a contradiction? I thought $\frac{0}{0}$ was undefined.

Find the value of $$S=\sum_{n=1}^{\infty}\left(\frac{2}{n}-\frac{4}{2n+1}\right)$$ My Try:we have $$S=2\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{2}{2n+1}\right)$$ $$S=2\left(1-\frac{2}{3}+\frac{1}{2}-\frac{2}{5}+\frac{1}{3}-\frac{2}{7}+\cdots\right)$$ so $$S=2\left(1+\frac{1}{2}-\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\cdots\right)$$ But we know $$\ln2=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\cdots$$ So $$S=2(2-\ln 2)$$ Is this correct?

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