I'm new in calculus and I couldn't solve this equation

Question

$|A - B| = \sqrt{2}|A + B|$ this is the equation and the question is, what's the magnitude of $B$?

Answer 1

HINT

Note that $A=0 \iff B=0$ is a trivial solution, then assume $A\neq 0$ and let $x=B/A$ then

$$|A - B| = \sqrt{2}|A + B|\iff |1 - x| = \sqrt{2}|1 + x|$$

hence we need to consider three cases

• $x<-1 \implies x-1=\sqrt 2(-1-x)$

• $-1\le x<1 \implies 1-x=\sqrt 2(-1-x)$

• $x\ge 1 \implies 1-x=\sqrt 2(1+x)$