\Rightarrow (x^2 + y^2 + z^2) - 2x - 2z + 2 = 2

Optimize Your Equation: Solve and Simplify » (𝑥² + likewise for y² + z²) – A Step-by-Step Guide

If you’ve ever come across the equation  (𝑥² + y² + z²) – 2𝑥 – 2𝑧 + 2 = 2, you're not alone. This mathematical expression combines algebraic manipulation and geometric interpretation, making it a great example for learners, students, and anyone exploring quadratic surfaces. Today, we’ll break down how to simplify and interpret this equation — turning it into a clearer, actionable form.

Understanding the Context

Understanding the Equation

The given equation is:

(𝑥² + y² + z²) – 2𝑥 – 2z + 2 = 2

At first glance, it’s a quadratic in three variables, but notice how several terms resemble the expansion of a squared binomial. This realization is key to simplifying and solving it.

$\Rightarrow (x^2 + y^2 + z^2) - 2x - 2z + 2 = 2$

Key Insights

Step 1: Simplify Both Sides

Subtract 2 from both sides to simplify:

(𝑥² + y² + z²) – 2𝑥 – 2z + 2 – 2 = 0

Simplify:

𝑥² + y² + z² – 2𝑥 – 2z = 0

🔗 Related Articles You Might Like:

📰 You Won’t Believe How Long Pizza Stays Fresh in the Fridge—This 4-Part Breakdown Will Surprise You! 📰 Don’t Waste Your Pizza! Here’s Exactly How Long It Lasts in the Fridge (Fact Check Inside!) 📰 Pizza in the Fridge? Here’s the Shocking Truth About Fridge Freshness—Don’t Guess, Know! 📰 This Full Size Keyboard Just Broke The Internetdont Miss Out On The Trending Key Choice 📰 This Full Size Trundle Bed Hides Space Like A Secret Fortyoull Never Guess How Space Saving It Truly Is 📰 This Fullmetal Alchemist Revelation Will Change How You Watch Every Episode Forever 📰 This Fully Grown Cavapoochon Is Taking The Internet By Stormare You Ready 📰 This Fumanchu Song Will Get You Dancing Tonight Heres Why Its The Hit Of The Week 📰 This Fun Of Games Secret Will Have You Playing All Night Longdont Miss It 📰 This Funeral Program Revealed Lifes Greatest Lessons You Havent Heard 📰 This Funimation Anime Twist Is Taking Youtube By Stormwatch Now 📰 This Funny Bunny Nail Polish Will Have You Giggling Unlock The Cutest Trend 📰 This Funny Face Meme Is Making Everyone Laugh Can You Guess The Origin 📰 This Funny Valentine Recipe Will Make Your Claim Heart Feltand Make Everyone Laugh 📰 This Fur Blazer Men Looks Too Expensiveyou Wont Believe How It Transforms Your Style 📰 This Fur Coat Jacket For Women Will Make You The Most Stylish Renegade This Season 📰 This Fur Elise Partitura Reveal Will Change How You Play Classical Music Forever 📰 This Fur Hat Is Selling Faster Than Everdont Miss Out

Final Thoughts

Step 2: Complete the Square

Group the terms involving 𝑥 and z to complete the square:

For 𝑥² – 2𝑥:
𝑥² – 2𝑥 = (𝑥 – 1)² – 1
For z² – 2z:
z² – 2z = (z – 1)² – 1

Now substitute back:

(𝑥 – 1)² – 1 + y² + (z – 1)² – 1 = 0

Combine constants:

(𝑥 – 1)² + y² + (z – 1)² – 2 = 0

Move constant to the right:

(𝑥 – 1)² + y² + (z – 1)² = 2