Articles of algebra precalculus

Prove that there exist infinitely many pythagorean integers $a²+b²=c²$

Prove that there exist infinitely many Pythagorean integers $a²+b²=c²$ My key idea is to show that there exists infinitely many integers that can be the length of the sides of a right triangle, but I fail at it. Other try is that $\sqrt{a^2+b^2}=c$ and so it is an equation of a circle, so I tried […]

How can we solve: $\sqrt{x} + \ln(x) -1 = 0$?

How could we solve $$\sqrt{x} + \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions?

Finding roots of polynomials, negative square root

The formula for finding the roots of a polynomial is as follows $$x = \frac {-b \pm \sqrt{ b^2 – 4ac }}{2a} $$ what happens if you want to find the roots of a polynomial like this simplified one $$ 3x^2 + x + 24 = 0 $$ then the square root value becomes $$ […]

Can a fourth-order equation be solved like a quadratic equation?

I was asked to find the zeros of $y = x^4 + 5x^2 +6$. I tried to turn this into a quadratic to factor it as follows: $y = x^4 + 5x^2 +6 = {(x^2)}^2 + 5{(x^2)}^1 + 6$ Put another way: Let $t = {x^2}$ $y = t^2 + 5t +6$ $y = (t […]

Determine limit of |a+b|

This is a simple problem I am having a bit of trouble with. I am not sure where this leads. Given that $\vec a = \begin{pmatrix}4\\-3\end{pmatrix}$ and $|\vec b|$ = 3, determine the limits between which $|\vec a + \vec b|$ must lie. Let, $\vec b = \begin{pmatrix}\lambda\\\mu\end{pmatrix}$, such that $\lambda^2 + \mu^2 = 9$ […]

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 […]