DMCA
Contact
Privacy Policy

Home » The Last Man Brian Vaughan Exposes the Shocking Truth That Changed Comics Forever! - ECD Germany

The Last Man Brian Vaughan Exposes the Shocking Truth That Changed Comics Forever! - ECD Germany

📅 May 9, 2026 👤 scraface

May 09, 2026

Content is being prepared. Please check back later.

📚 You May Also Like These Articles

📌 $ 4(x+3)^2 - 36 - 9(y-1)^2 + 9 = -9 $.

📌 $ 4(x+3)^2 - 9(y-1)^2 = 18 $.

📌 Divide by 18:

📌 $ rac{(x+3)^2}{4.5} - rac{(y-1)^2}{2} = 1 $.

📌 Center at $ (-3, 1) $. Final answer: oxed{(-3,\ 1)}

📌 Question: Let $ z $ and $ w $ be complex numbers such that $ z + w = 2 + 4i $ and $ z \cdot w = 13 - 2i $. Find $ |z|^2 + |w|^2 $.

📌 Solution: Use $ |z|^2 + |w|^2 = |z + w|^2 - 2 ext{Re}(z \overline{w}) $. Compute $ |z + w|^2 = |2 + 4i|^2 = 4 + 16 = 20 $. Let $ z \overline{w} = a + bi $, then $ ext{Re}(z \overline{w}) = a $. From $ z + w = 2 + 4i $ and $ zw = 13 - 2i $, note $ |z|^2 + |w|^2 = (z + w)(\overline{z} + \overline{w}) - 2 ext{Re}(z \overline{w}) = |2 + 4i|^2 - 2a = 20 - 2a $. Also, $ zw + \overline{zw} = 2 ext{Re}(zw) = 26 $, but this path is complex. Alternatively, solve for $ |z|^2 + |w|^2 = |z + w|^2 - 2 ext{Re}(z \overline{w}) $. However, using $ |z|^2 + |w|^2 = (z + w)(\overline{z} + \overline{w}) - 2 ext{Re}(z \overline{w}) = |z + w|^2 - 2 ext{Re}(z \overline{w}) $. Since $ z \overline{w} + \overline{z} w = 2 ext{Re}(z \overline{w}) $, and $ (z + w)(\overline{z} + \overline{w}) = |z|^2 + |w|^2 + z \overline{w} + \overline{z} w = |z|^2 + |w|^2 + 2 ext{Re}(z \overline{w}) $, let $ S = |z|^2 + |w|^2 $, then $ 20 = S + 2 ext{Re}(z \overline{w}) $. From $ zw = 13 - 2i $, take modulus squared: $ |zw|^2 = 169 + 4 = 173 = |z|^2 |w|^2 $. Let $ |z|^2 = A $, $ |w|^2 = B $, then $ A + B = S $, $ AB = 173 $. Also, $ S = 20 - 2 ext{Re}(z \overline{w}) $. This system is complex; instead, assume $ z $ and $ w $ are roots of $ x^2 - (2 + 4i)x + (13 - 2i) = 0 $. Compute discriminant $ D = (2 + 4i)^2 - 4(13 - 2i) = 4 + 16i - 16 - 52 + 8i = -64 + 24i $. This is messy. Alternatively, use $ |z|^2 + |w|^2 = |z + w|^2 + |z - w|^2 - 2|z \overline{w}| $, but no. Correct approach: $ |z|^2 + |w|^2 = (z + w)(\overline{z} + \overline{w}) - 2 ext{Re}(z \overline{w}) = 20 - 2 ext{Re}(z \overline{w}) $. From $ z + w = 2 + 4i $, $ zw = 13 - 2i $, compute $ z \overline{w} + \overline{z} w = 2 ext{Re}(z \overline{w}) $. But $ (z + w)(\overline{z} + \overline{w}) = 20 = |z|^2 + |w|^2 + z \overline{w} + \overline{z} w = S + 2 ext{Re}(z \overline{w}) $. Let $ S = |z|^2 + |w|^2 $, $ T = ext{Re}(z \overline{w}) $. Then $ S + 2T = 20 $. Also, $ |z \overline{w}| = |z||w| $. From $ |z||w| = \sqrt{173} $, but $ T = ext{Re}(z \overline{w}) $. However, without more info, this is incomplete. Re-evaluate: Use $ |z|^2 + |w|^2 = |z + w|^2 - 2 ext{Re}(z \overline{w}) $, and $ ext{Re}(z \overline{w}) = ext{Re}(rac{zw}{w \overline{w}} \cdot \overline{w}^2) $, too complex. Instead, assume $ z $ and $ w $ are conjugates, but $ z + w = 2 + 4i $ implies $ z = a + bi $, $ w = a - bi $, then $ 2a = 2 \Rightarrow a = 1 $, $ 2b = 4i \Rightarrow b = 2 $, but $ zw = a^2 + b^2 = 1 + 4 = 5

📌 eq 13 - 2i $. So not conjugates. Correct method: Let $ z = x + yi $, $ w = u + vi $. Then:

🔥 Popular Posts

smokey and the bandit cast
cast of the snitch
jumped a shark
blyth dolores barrymore
robert crawford
lincoln lawyer season 2
barbara hale actress
versailles tv series cast
deirdre o connell
bryan brown movies and tv shows

📝 Recent Posts

Master How to Breed Villagers Instantly – Ultimate Minecraft Breeding...
No More Waiting! Discover the Fastest Way to Breed Minion Villagers in...
Unlock the Best Villager Breeding Strategies That Every Minecraft Play...
Finally! The Proven Method to Breed Hundreds of Villagers Overnight in...
GET RID OF CHEAT DETECTORS: THE SHOCKING SECRET TO ALLOWING CHEATS IN...
YOU’LL NEVER BELIEVE HOW EASY IT IS TO LET CHEATS WORK IN MINECRAFT –...
HACK MINECRAFT LIKE A CHAMP: THE FORBIDDEN GUIDE TO TURNING CHEATS ON...
STOP BEING PUNISHED: UNLOCK CHEAT CODES & LOOK SEMI-INVINCIBLE IN Mine...
INSIDER TRICK: ALLOW CHEATS IN MINECRAFT WITHOUT GLITCHES – YOUR NEW S...
This Simple Secrets to Brine a Turkey and Get a Juicy Thanksgiving Fea...

© 2026 ECD Germany · Powered by WordPress

Disclaimer: This site is an informational archive compiling publicly available global death and memorial records. It is not a news agency, media organization, nor an entertainment platform.