How-many-distinct-y-exist-satisfying-x-2-8x-18-y-3-5-

Question Number 176774 by Ar Brandon last updated on 26/Sep/22

How many distinct y exist satisfying

∣ x^{2} - 8 x + 18 ∣ + ∣ y - 3 ∣= 5

Answered by mr W last updated on 26/Sep/22

x, y \in Z

∣ {(x - 4)}^{2} + 2 ∣ + ∣ y - 3 ∣= 5

2 + ∣ y - 3 ∣= 5 \Rightarrow y - 3 = \pm 3 \Rightarrow y = 0, 6

3 + ∣ y - 3 ∣= 5 \Rightarrow y - 3 = \pm 2 \Rightarrow y = 1, 5

4 d i c t i n c t v a l u e s o f y e x i s t .

Commented by Ar Brandon last updated on 27/Sep/22

Thank you Sir. I now get it.

Commented by mr W last updated on 27/Sep/22

∣ {(x - 4)}^{2} + 2 ∣ + ∣ y - 3 ∣= 5

l e t t = x - 4

∣ t^{2} + 2 ∣ + ∣ y - 3 ∣= 5

t^{2} c a n o n l y b e 0, 1, 4, 9, 16 e t c .

∣ t^{2} + 2 ∣ c a n o n l y b e 2, 3, 6, 11, 18 e t c .

w i t h ∣ t^{2} + 2 ∣= 2 y o u g e t 2 + ∣ y - 3 ∣= 5 .

w i t h ∣ t^{2} + 2 ∣= 3 y o u g e t 3 + ∣ y - 3 ∣= 5 .

w i t h ∣ t^{2} + 2 ∣= 6 y o u g e t 6 + ∣ y - 3 ∣= 5 \Rightarrow n o s o l u t i o n!

Commented by Ar Brandon last updated on 26/Sep/22

Great! Thank you, Sir!

Commented by Ar Brandon last updated on 27/Sep/22

Sir why

2 + ∣ y - 3 ∣ and 3 + ∣ y - 3 ∣ ?