\Rightarrow (x^2 + y^2 + z^2) - 2x - 2z + 2 = 2 - Crosslake
Optimize Your Equation: Solve and Simplify » (𝑥² + likewise for y² + z²) – A Step-by-Step Guide
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.
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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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
