Articles of algebra precalculus

Algorithm/Formula to compute adding and/or removing compound and/or non-compound percentages from a value?

I will first start with a scenario, I have to apply some adjustments to a particular value. These adjustments are either compound or non-compounded and they can either be added or subtracted to the value. They are executed in the order they are stored (they are stored in a database with a sequence number starting […]

Perpendicular Bisector of Made from Two Points

For a National Board Exam Review: Find the equation of the perpendicular bisector of the line joining (4,0) and (-6, -3) Answer is 20x + 6y + 29 = 0 I dont know where I went wrong. This is supposed to be very easy: Find slope between two points: $${ m=\frac{y^2 – y^1}{ x^2 – […]

Solving 3 simultaneous cubic equations

I have three equations of the form: $$i_1^3L_1+i_1K+V_1+(i_2+i_3+C)Z_n=0$$ $$i_2^3L_2+i_2K+V_2+(i_1+i_3+C)Z_n=0$$ $$i_3^3L_3+i_3K+V_3+(i_1+i_2+C)Z_n=0$$ where $L_1,L_2,L_3,K,V_1,V_2,V_3,C$ and $Z_n$ are all known constants. What methods can I use to obtain the values of $i_1,i_2$ and $i_3$ ?

Non-integer exponents

Can you use noninteger powers Like is $x^{8.3} / x^{2.2} = x^{6.1}$?

Show that $\tan {\pi \over 8} = \sqrt 2 – 1$

Show that $\tan {\pi \over 8} = \sqrt 2 – 1$, using the identity $\tan 2\theta = {{2\tan \theta } \over {1 – {{\tan }^2}\theta }}$ Using $\tan 2\theta = {{2\tan \theta } \over {1 – {{\tan }^2}\theta }}$ with $\theta = {\pi \over {16}}$: $\eqalign{ & \tan {\pi \over 8} = {{2\tan {\pi \over […]

Fundamental period of two functions

I have the function $$f(t)=\sin(t)+\sin(\sqrt2t)$$ I would like to calculate the fundamental period of $f(t)$. I know that the period of $\sin(t)$ is $2\pi$ and the period of $\sin(\sqrt2t)$ is $\sqrt2\pi$. I sense that I must work out the lcm of $2$ and and $\sqrt2$ but I’m unsure on how to do this.

What is an easy way to prove the equality $r > s > 0$ implies $x^r > x^s$?

I have been using simple inequalities of fractional powers on a positive interval and keep abusing the inequality for $x>1$. I was just wondering if there is a nice way to prove the inequality in a couple line: Let $x \in [1,\infty)$ and $r,s \in \mathbb{R}$ What is an easy way to prove the equality […]

Complete the squares to find the center and radius of the circle

I’ve had trouble on this one problem for a couple days. Complete the square on the X and Y terms to find the center and radius of the circle. $x^2+2x+y^2-4y=-4\:\:$

Rotation of complex numbers in a complex plane. Check my work?

Say that $c_1 = -i$ and $c_2 = 3$. For this problem, let $z_0$ be an arbitrary complex number. We can rotate $z_0$ around $c_1$ by $\pi/4$ counterclockwise to get $z_1$. Next, we canrotate $z_1$ around $c_2$ by $\pi/4$ counter-clockwise to get $z_2$. There exists a complex number $c$ where we can get $z_2$ from […]

Minimum of $\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}$

Let $0\leq x,y,z\leq 1$. What is the minimum of $$F(x,y,z)=\frac{1}{x+y}+\frac{1}{x+z}-\frac{1}{x+y+z}?$$ We have $F(1,1,1)=2/3$, $F(1,1,0)=F(1,0,1)=F(1,0,0)=1$, and $F(0,1,1)=3/2$. Is the minimum $2/3$?