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

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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

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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