2a x^2 + 2a y^2 + 2b x + 2c = 2a x^2 + 2a y^2 + 2b x + 2b y + 4c.
Related Articles
- 5Question: Find all functions $ f: \mathbb{R} o \mathbb{R} $ such that $ f(x + y) + f(x - y) = 2f(x) + 2f(y) $ for all real numbers $ x $ and $ y $.
- Solution: Assume $ f $ is a quadratic function. Let $ f(x) = ax^2 + bx + c $. Substituting into the equation:
- a(x + y)^2 + b(x + y) + c + a(x - y)^2 + b(x - y) + c = 2(ax^2 + bx + c) + 2(ay^2 + by + c).
- Equating coefficients: $ 2c = 2b y + 4c $ for all $ y $, which implies $ b = 0 $ and $ c = 0 $. Thus, $ f(x) = ax^2 $. Verification confirms this satisfies the equation. Hence, all solutions are quadratic functions of the form $ f(x) = kx^2 $.
- oxed{f(x) = kx^2 ext{ for some constant } k}
- Question: Let $ a $ and $ b $ be complex numbers such that $ |a| = |b| = 1 $ and $ a
Trending Articles
The Secret Spell That Turns Magicka Against Your Enemies
Boss Fight Salad: This Strawberry Recipe is Your Ultimate Cheat Code
Why Does Elden Ring Hide This Desktop Icon Secret?
Did You See The November Axolotl Glitch In Fallout 76 Yet?
Plants vs Zombies Replanted Preorder Bonuses: Unlock the Secret Skin Before Launch
Rivals 2 Workshop Best Community Maps Ranked
The Cat Dealing a Royal Flush? Watch This!
Secrets Rock Roll Racing Hides That Devs Don't Want You to See
Why Are Gamers Obsessed With This One Weird Trick?
Player Chess Challenge: The Addictive Game You Never Knew You Needed