IrishBoy123
  • IrishBoy123
complex number
Mathematics
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IrishBoy123
  • IrishBoy123
complex number
Mathematics
schrodinger
  • schrodinger
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
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IrishBoy123
  • IrishBoy123
for \( |z| = 2\), prove: \[|z^2 - 4 z - 3 | \le 15\]
freckles
  • freckles
\[-15 \le z^2-4z-3 \le 15 \\ -15+3 \le z^2-4z \le 15+3 \\ -12 \le z^2-4z \le 18 \\ -12+4 \le z^2-4z+4 \le 18+4 \\ -8 \le (z-2)^2 \le 22 \\ 0 \le (z-2)^2 \le 22 \\ (z-2)^2 \le 22 \\ -\sqrt{22} \le z-2 \le \sqrt{22} \\ \] I wonder if it has anything to do with this I'm still thinking
Loser66
  • Loser66
I wonder if I take it easy?? Using Triangle in equality, I have \(|z^2-4z -3|\leq |z^2| +|-4z|+|-3|= |z|^2 +4|z| +|-3)= 2^2 +4*2 +3 =15\)

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freckles
  • freckles
I think loser's way works awesome
Loser66
  • Loser66
Thank you. I was doubt myself.
IrishBoy123
  • IrishBoy123
@Loser66 which are the sides of the triangle??
IrishBoy123
  • IrishBoy123
can you draw it pls?
Loser66
  • Loser66
|dw:1440896633171:dw|
Loser66
  • Loser66
That is the triangle inequality theorem.
Loser66
  • Loser66
\(|x+y|\leq |x| +|y|\)
IrishBoy123
  • IrishBoy123
yes, i can see that, it is absolutely marvellous however, i cannot see how we apply it to \[z^2āˆ’4zāˆ’3\]
freckles
  • freckles
\[|z^2-4z-3|=|z^2+(-4z-3)| \le |z^2|+|(-4z-3)| =|z|^2+|(-1)(4z+3)| \\ =2^2+|-1| \cdot |4z+3|=4+1 \cdot |4z+3| \\= 4+|4z+3| \le 4+|4z|+|3| =4+|4| \cdot |z|+3 \\ =4+4(2)+3=4+8+3=15\] is it the three term thing @IrishBoy123 ?
Loser66
  • Loser66
Yes! same. :)
freckles
  • freckles
http://mathworld.wolfram.com/TriangleInequality.html triangle inequality can hold for more than two term
IrishBoy123
  • IrishBoy123
yep @freckles it is the 3 term thing i can see the basic triangle thing but i am staring at a quadratic in z and it is not so clear
freckles
  • freckles
Well you could apply the triangle inequality twice as I did above
freckles
  • freckles
|dw:1440897294917:dw|
IrishBoy123
  • IrishBoy123
right, i need to read some more brilliant, @freckles and @Loser66 much appreciated! i will revert :p
IrishBoy123
  • IrishBoy123
and by read, i mean read 'this thread'
freckles
  • freckles
loser was the brilliant one he was like oh lets use triangle inequality
freckles
  • freckles
oops she
freckles
  • freckles
I'm so sorry loser
IrishBoy123
  • IrishBoy123
"I'm so sorry loser" you could not make this up! good night lovely people! sleep well!
IrishBoy123
  • IrishBoy123
well, depends on what time zone you are in, i suppose, but......zzzzzzzzz.
freckles
  • freckles
goodnight
IrishBoy123
  • IrishBoy123
got there! posting a related question. i think i know how to do it now but i am interested in seeing what people have to say.

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