+0

Help algebra

+1
97
1

$x = 1 + \cfrac{\sqrt{3}}{1 + \cfrac{\sqrt{3}}{1 + \dotsb}}$.  Find $\frac{1}{(x + 1)(x - 2)}$.  When your answer is in the form $\frac{A + \sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, and $B$ is not divisible by the square of a prime, what is $|A| + |B| + |C|$?

Jun 15, 2021

#1
+449
+1

First, simplify the first equation:

$$x=1+\frac{\sqrt{3}}{x}\\x^2-x=\sqrt{3}$$

Then, notice that $$\frac{1}{(x+1)(x-2)}$$ simplifies to $$\frac{1}{x^2-x-2}$$, so we can replace it like so:

$$\frac{1}{\sqrt{3}-2}$$

To simplify, just multiply the top and bottom by the conjugate

$$\frac{\sqrt{3}+2}{(\sqrt{3}+2)(\sqrt{3}-2)}\\=\frac{\sqrt{3}+2}{3-4}=\frac{2+\sqrt{3}}{-1}$$

$$|2|+|3|+|-1|=\boxed{6}$$

Jun 15, 2021