Articles of algebra precalculus

Solutions to exp(x) + x = 2

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 […]

Pi Estimation using Integers

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 […]

Is there a general formula for solving 4th degree equations (quartic)?

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?

Why rationalize the denominator?

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 $\frac{1}{\sqrt{2}}$ – […]

Pedagogy: How to cure students of the “law of universal linearity”?

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 […]

Find $\frac{a^2}{2a^2+bc}+\frac{b^2}{2b^2+ac}+\frac{c^2}{2c^2+ab}$ if $a+b+c=0$

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 […]

Derivation of the formula for the vertex of a parabola

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 […]

How do I start from a 10% discount and find the original price?

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 this number fact

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 $\sum_{n=1}^{\infty} \frac{2}{n}-\frac{4}{2n+1}$

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?