Articles of algebra precalculus

Next number in series

What are the basic/advanced strategies used to find the next number in series. I know the simple ones such as addition, multiplication etc. But recently I came into a question that goes on something like 812, 819, 823, 835, 834, 851(Don’t try to solve this, I changed some numbers and there is no sequence). This […]

How can I solve $\sin(x)=\sin(2x)$?

Solve $\sin(x)=\sin(2x)$ How do I solve this equation for $x$ without a calculator? It seems pretty simple but I’m not sure how to do it.

License plate consisting of 4 letters and 4 numbers

While doing homework today, the following question popped into my head: Can you easily calculate the amount of unique license plates consisting of 4 letters and 4 numbers in any order? It doesn’t seem to be easily possibly; $$ 26^3 * 10^3 * 8! $$ would include repeats (such as AAA123 and AAA123; As are […]

Factoring Cubic Equations

I’ve been trying to figure out how to factor cubic equations by studying a few worksheets online such as the one here and was wondering is there any generalized way of factoring these types of equations or do we just need to remember a bunch of different cases. For example, how would you factor the […]

Alternating sum of binomial coefficients is equal to zero

This question already has an answer here: Alternating sum of binomial coefficients: given $n \in \mathbb N$, prove $\sum^n_{k=0}(-1)^k {n \choose k} = 0$ 7 answers

Reference books for highschool Algebra and Geometry?

I’m tutoring high school students in Math for a local College and Career prep program and would like to have a reference book on hand that I can consult. I’m a Comp Sci graduate so I have a pretty strong background in Math but it’s been a while since I used high school level Algebra […]

Rationalizing the denominator of $\frac {\sqrt{10}}{\sqrt{5} -2}$

I have the expression $$\frac {\sqrt{10}}{\sqrt{5} -2}$$ I can’t figure out what to do from here, I can’t seem to pull any numbers out of either of the square roots so it appears that it must remain as is.

The Roots of Unity and the diagonals of the n-gon inscribed in the unit circle

I want to prove that the sum of the squares of the diagonals of a regular $n$-gon inscribed in the unit circle is equal to $2n$. So what I’ve done is I considered the $n$th roots of unity and said that the diagonals are given by $$|1-\omega^k|$$ where $k=1,2,3,…,(n-1)$. Then the sum of the squares […]

Solving $\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$, $\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$

Whether non-zero integers $a, b, c$ with the property that $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$$ and $$\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$$ Calculate all possible values for $m + n$.

Proportional Distribution

I have a problem regarding supply distribution. I distribute widgets on a monthly basis; I have many customers and each of them request a different quantity each month. My monthly supply is limited and I cannot fill every order. How do I distribute fairly, across the board? There must be some sort proportional way to […]