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.

Solved 1. y=x^2 - z2 2. x^2 - y^2 + z^2 = 1 3. y = 2x^2 + | Chegg.com

Image Gallery

Solved 1. y=x^2 - z2 2. x^2 - y^2 + z^2 = 1 3. y = 2x^2 + | Chegg.com
Solved z^2 = x^2 + y^2 _____ z^2 = -1 + x^2 + y^2 _____ | Chegg.com
Solved x2+y2=1 x2+y2+z2=1 z=x+y z=x2−y2 z=x2 z2=x2+y2 | Chegg.com
Solved (a) z=2+x2+y2 (b) z=2+x−2y (c) z=2−x+2y (d) | Chegg.com
Solved x2+y2+z2=2x+2y+2z−2 | Chegg.com

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

Continue Reading

‘Marked for Death’—The Chilling Truth Behind This Unsettling Sign!
steht on the Edge of Elimination—Marked for Death, Authority Says So!
Marlboro Coupons: Unlock $50 Off Your Next Cigarette Order – Act Now!

🔗 Related Articles You Might Like:

📰 Bank of America Lincoln 📰 Bank of America Pittsburg California 📰 15 Yr Refi Rates 📰 Thedailyrecord Wooster Ohio 📰 Finally A Simple Way To Master Oracle Bi Discoverers Powerful Features 3015186 📰 Your Sweetest Dreams Were Woven From These Lyrics 8810743 📰 Tabletop Games That Are Taking The World By Stormdont Miss The Top Picks 5548725 📰 Marilyn Mansion 5834812 📰 Bank Of America On Kirby 📰 Wells Fargo Bank Mobile Banking 📰 List Of Omega Mutants 8340691 📰 Number For Verizon Cable 📰 St Johns Mens Basketball Schedule 9328718 📰 Here Is The List Of Five Clickbait Titles For How Much Does 5 Gallons Of Water Weigh 8509561 📰 You Wont Believe How Easy It Is To Access Hhs Contact Infoclick Now 1245562 📰 X Frac 8 Pm Sqrt162 Times 2 Frac8 Pm 44 5239872 📰 Tmz Stock Shocking Surge You Wont Believe Whats Happening Right Now 5743272 📰 Face Off Streaming

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