## IrishBoy123 one year ago complex number

1. IrishBoy123

for $$|z| = 2$$, prove: $|z^2 - 4 z - 3 | \le 15$

2. 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

3. Loser66

I wonder if I take it easy?? Using Triangle in equality, I have $$|z^2-4z -3|\leq |z^2| +|-4z|+|-3|= |z|^2 +4|z| +|-3)= 2^2 +4*2 +3 =15$$

4. freckles

I think loser's way works awesome

5. Loser66

Thank you. I was doubt myself.

6. IrishBoy123

@Loser66 which are the sides of the triangle??

7. IrishBoy123

can you draw it pls?

8. Loser66

|dw:1440896633171:dw|

9. Loser66

That is the triangle inequality theorem.

10. Loser66

$$|x+y|\leq |x| +|y|$$

11. IrishBoy123

yes, i can see that, it is absolutely marvellous however, i cannot see how we apply it to $z^2−4z−3$

12. 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 ?

13. Loser66

Yes! same. :)

14. freckles

http://mathworld.wolfram.com/TriangleInequality.html triangle inequality can hold for more than two term

15. 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

16. freckles

Well you could apply the triangle inequality twice as I did above

17. freckles

|dw:1440897294917:dw|

18. IrishBoy123

right, i need to read some more brilliant, @freckles and @Loser66 much appreciated! i will revert :p

19. IrishBoy123

20. freckles

loser was the brilliant one he was like oh lets use triangle inequality

21. freckles

oops she

22. freckles

I'm so sorry loser

23. IrishBoy123

"I'm so sorry loser" you could not make this up! good night lovely people! sleep well!

24. IrishBoy123

well, depends on what time zone you are in, i suppose, but......zzzzzzzzz.

25. freckles

goodnight

26. 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.