Optimize Your Equation: Solve and Simplify Β»β€―(π‘₯Β² + likewise for yΒ² + zΒ²) – A Step-by-Step Guide

If you’ve ever come across the equation  (π‘₯Β² + yΒ² + zΒ²) – 2π‘₯ – 2𝑧 + 2β€―=β€―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 β€” turning it into a clearer, actionable form.

Understanding the Context

Understanding the Equation

The given equation is:

(π‘₯Β² + yΒ² + zΒ²) – 2π‘₯ – 2z + 2β€―=β€―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=x2βˆ’y2 z=x2 z2=x2+y2 | Chegg.com
Solved (a) z=2+x2+y2 (b) z=2+xβˆ’2y (c) z=2βˆ’x+2y (d) | Chegg.com
Solved x2+y2+z2=2x+2y+2zβˆ’2 | Chegg.com

Key Insights

Step 1: Simplify Both Sides

Subtract 2 from both sides to simplify:

(π‘₯Β² + yΒ² + zΒ²) – 2π‘₯ – 2z + 2 – 2β€―=β€―0

Simplify:

π‘₯Β² + yΒ² + zΒ² – 2π‘₯ – 2zβ€―=β€―0

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Final Thoughts

Step 2: Complete the Square

Group the terms involving π‘₯ and z to complete the square:

  • For π‘₯Β² – 2π‘₯:
    π‘₯Β² – 2π‘₯ = (π‘₯ – 1)Β² – 1
  • For zΒ² – 2z:
    zΒ² – 2z = (z – 1)Β² – 1

Now substitute back:

(π‘₯ – 1)Β² – 1 + yΒ² + (z – 1)Β² – 1 = 0

Combine constants:

(π‘₯ – 1)Β² + yΒ² + (z – 1)Β² – 2 = 0

Move constant to the right:

(π‘₯ – 1)Β² + yΒ² + (z – 1)Β² = 2