\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 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.
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
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
Step 3: Interpret the Equation
This final form reveals a sphere in 3D space:
- Center: (1, 0, 1)
- Radius: √2
In geometric terms, the original equation defines all points (𝑥, y, z) that lie on the surface of a sphere centered at (1, 0, 1) with radius √2.
Why This Matters and How to Use It
Understanding equations like this appears in fields such as:
- 3D geometry and computer graphics: modeling spheres and surfaces.
- Optimization problems: minimizing or maximizing functions constrained by geometric shapes.
- Physics and engineering: representing spatial boundaries or constraints.
By completing the square, we transformed a complex-looking quadratic expression into a clear geometric representation.
Summary
- Original Eq: (𝑥² + y² + z²) – 2𝑥 – 2z + 2 = 2
- Simplified: (𝑥 – 1)² + y² + (z – 1)² = 2
- Geometric Interpretation: Sphere centered at (1, 0, 1) with radius √2
Final Thoughts
Solving equations is more than simplifying symbols — it’s uncovering hidden structure. This example illustrates how algebraic manipulation reveals geometric insight. Whether you're solving equations for homework, coding a simulation, or exploring math visually, mastering these techniques empowers you to better understand spatial relationships and optimize complex systems.
Try it yourself: Plot this sphere on graphing tools or experiment with transforming other quadratic forms — you’ll unlock deeper insights every time.
If you want to dive deeper into solving 3D quadratic equations or visualizing surfaces, stay tuned — more math simplifications await!
