Articles of algebra precalculus

Why does $\sqrt{1/2} = 1/\sqrt{2}$ but $\sqrt{2/3} \neq 2/\sqrt{3}$?

I can see that $\sqrt{1/2} = 1/\sqrt{2}$ My calculator also confirms I can change the denominator and the equality still holds. But $\sqrt{2/3} \neq 2/\sqrt{3}$ Can someone explain why? I need to get the concept here.

Stuck on rearranging of this equation

I need to get from $[(1-p)f+p(1-f)](1+v)-[(1-p)(1-f)+pf] = x$ to $(2+v)(f+p-2pf)-1 = x$ but I’m stuck. I’d appreciate any tips on what I should I do after the following. $(f+p-2pf)(1+v) + (f + p – 2pf) – 1 = x$ Thanks in advance.

Quadratic equation which has rational roots

If the following quadratic equation $$qx^2+(p+q)x+bp=0$$ always has rational roots for any non-zero integers $p$ and $q$ what will be the value of $b$? My book’s solution says the value of $b$ will be $0$ or $1$. If we consider the discriminant of the equation, $$D=(p+q)^2-4bqp = p^2+2q(1-2b)p+q^2$$ then $D$ should be a perfect square […]

Absolute value of a real number

My question is: Solve: $|x-4|< a$, where $a$ belongs to the real numbers. Solve this by considering various cases depending upon whether $a$ is negative, positive or zero. What I have tried so far: If $a>0$ then: $x < a+4$ and $x>4-a$, if $a=0$ then there is no solution. My doubt is: Should I consider […]

Prove that $\frac{bc}{b+c}+\frac{ac}{a+c}+\frac{ab}{a+b} \leq \frac{a+b+c}{2}$

If $a,b,c \in R$, then prove that: $$\frac{bc}{b+c}+\frac{ac}{a+c}+\frac{ab}{a+b} \leq \frac{a+b+c}{2}$$ I can’t see any known inequality working here like $H.M.-A.M.$. Could this be solved using basic inequalities?

Partial fractions of $\frac{-5x+19}{(x-1/2)(x+1/3)}$

Alright, I need to find the partial fractions for the expression above. I have tried writing this as $$\frac{a}{x-1/2}+\frac{b}{x+1/3}$$ but the results give me $a=25.8$ and $b=-20.8$, which are slightly wrong because they give me $5x+19$ instead of $-5x+19$. Can you please help? Thanks a lot

How can I solve for a single variable which occurs in multiple trigonometric functions in an equation?

This is a pretty dumb question, but it’s been a while since I had to do math like this and it’s escaping me at the moment (actually, I’m not sure I ever knew how to do this. I remember the basic trigonometric identities, but not anything like this). I have a simple equation of one […]

Finding solution with Lambert function

I have following equation to solve for $x$ $$\ln\left(1+\frac{bx}{a}\right)=\frac{4cx}{a}$$ where $a>0,b>0$ and $c>0$. In my own attempt I replaced $1+\frac{bx}{a}$ by $y$ and with this replacement the final form of the equation is $$ye^{-\frac{4cy}{b}}=e^{-\frac{4c}{b}}$$ I don’t know how to proceed further. Any help in this regard will be much appreciated. BR Frank

normal distribution hazard rate increasing function

How to show this function is increasing convex function: Define $f(z)=\frac{T(z)}{g(z)}$, where $T(z)=\phi(z)-\alpha \phi(\frac{z}{\alpha})+z(\Phi(z)-\Phi(\frac{z}{\alpha}))\,,$ $g(z)=\Phi(z)-\frac{1}{2}\Phi(\frac{z}{\alpha})-\int_{-\infty}^{z}\phi(x)\Phi(\sqrt{\frac{1-\alpha^2}{\alpha^2}}\,x)) dx$ and $\Phi(z)=\int_{-\infty}^z\frac{1}{\sqrt{2\pi}}e^{-\frac{x^2}{2}}$ is the CDF of standard normal distribution and $\phi(z)=\frac{1}{\sqrt{2\pi}}e^{-\frac{z^2}{2}}$ is the PDF of standard normal distribution and $0<\alpha<1$. How can we show $f(z)$ is increasing and convex (numerical graph of the function is increasing and convex)? (maybe […]

$\forall m\in\mathbb N\exists n>m+1\exists N\in\mathbb N:m^2+n^2+(mn)^2=N^2$

Prove the conjecture or give a counter-example: For each $m\in\mathbb N$ there exist a $n>m+1$ such that $m^2+n^2+(mn)^2$ is a perfect square. I have just tried it out numerically and it holds for $m<1000$. I can’t see any pattern for the smallest $n$: m n 1 12 2 8 3 18 4 32 5 50 […]