Articles of algebra precalculus

Need help with a power equation

If $x^3 + \frac{1}{x^3} = 52$, then what is $x^2 + \frac{1}{x^2}$? I’m not sure which formulae or methods should I used to solve this problem, so could somebody show me a way? what should I be looking for?

Work out the values of a and b from the identity $x^2 – ax + 144 = (x-b)^2$

How do I solve the following question? You are given the identity $x^2-ax+144 = (x-b)^2$ Work out the values of $a$ and $b$. Question appears in AQA 43005/1H.

Value of $ \cos 52^{\circ} + \cos 68^{\circ} + \cos 172^{\circ} $?

I am a little weak in trigonometry. I have two questions: Find the value of $\cos 52^{\circ} + \cos 68^{\circ} + \cos 172^{\circ} $ Find the value of $\sin 28^{\circ}+ \cos 17^{\circ} + \cos 28^{\circ} + \sin 17^{\circ} $ I am asking these questions because: 1. I am weak and unable to solve these. 2. […]

Prove that if function f is monotonic, then it one-to-one

What I have so far: Suppose $f$ is monotonic. It is therefore either increasing or decreasing. Proof for increasing: If $f$ is increasing, then $f(x_1) <f(x_2)$ whenever $x_1 < x_2$, which means $f(x_1) = f(x_2)$ if, and only if, $x_1 = x_2$. Therefore $f$ is one to one. I think this is wrong though. Could […]

Whats the rule for putting up a plus-minus sign when taking under root?

Given a simple equation…. $\ (x+1)^2 =21 $ if we take the under root of both sides , we get $\ x+1 = \pm \sqrt{21} $ why dont we get a $ \pm $ on the left hand side ?

$n$-tuples of points of $\mathbb{C}$, identification.

Fix $n \in \mathbb{N}$. Forgive me if this is a very silly question, but how can I see that the set of unordered $n$-tuples of points of $\mathbb{C}$ can be naturally identified with $\mathbb{C}^n$?

Absolute Value inequality help: $|x+1| \geq 3$

Find the solutions to the inequality: $$|x+1| \geq 3$$ I translate this as: which numbers are at least $3$ units from $1$? So, picturing a number line, I would place a filled in circle at the point $1$. The solutions would then be on the interval $(-\infty,-2] \cup [4,\infty)$. But this is wrong, because: Why […]

Fractions with radicals in the denominator

I’m working my way through the videos on the Khan Academy, and have a hit a road block. I can’t understand why the following is true: $$\frac{6}{\quad\frac{6\sqrt{85}}{85}\quad} = \sqrt{85}$$

How to find logarithms of negative numbers?

Logarithms of negative numbers must be complex. But how do you find $\ln{(-2)}$ expressed in something like $x \cdot i$ where $x \in \mathbb{R}$?

Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$

Find minimum of $a+b+c+\frac1a+\frac1b+\frac1c$ given that: $a+b+c\le \frac32$ ($a,b,c$ are positive real numbers). There is a solution, which relies on guessing the minimum case happening at $a=b=c=\frac12$ and then applying AM-GM inequality,but what if one CANNOT guess that?!