simplify to sphere formula: y^2-10y+x^2+2x+z^2+35-6z=4y^2-16y+4x^2+4z^2+16z+176-48x

We are tasked with simplifying the given equation into the equation of a sphere. The equation is:

\[
y^2 - 10y + x^2 + 2x + z^2 + 35 - 6z = 4y^2 - 16y + 4x^2 + 4z^2 + 16z + 176 - 48x
\]

### Step 1: Simplify both sides
Start by moving all terms to one side of the equation:

\[
y^2 - 10y + x^2 + 2x + z^2 + 35 - 6z - \left( 4y^2 - 16y + 4x^2 + 4z^2 + 16z + 176 - 48x \right) = 0
\]

Distribute the negative sign on the right-hand side:

\[
y^2 - 10y + x^2 + 2x + z^2 + 35 - 6z - 4y^2 + 16y - 4x^2 - 4z^2 - 16z - 176 + 48x = 0
\]

### Step 2: Combine like terms
Now, combine the terms for \(y^2\), \(y\), \(x^2\), \(x\), \(z^2\), \(z\), and constants:

- \(y^2 - 4y^2 = -3y^2\)
- \(-10y + 16y = 6y\)
- \(x^2 - 4x^2 = -3x^2\)
- \(2x + 48x = 50x\)
- \(z^2 - 4z^2 = -3z^2\)
- \(-6z - 16z = -22z\)
- \(35 - 176 = -141\)

Thus, the equation simplifies to:

\[
-3y^2 + 6y - 3x^2 + 50x - 3z^2 - 22z - 141 = 0
\]

### Step 3: Divide through by -3
To simplify further, divide the entire equation by \(-3\):

\[
y^2 - 2y + x^2 - \frac{50}{3}x + z^2 + \frac{22}{3}z + 47 = 0
\]

### Step 4: Complete the square
To express the equation in the form of a sphere equation, we need to complete the square for the \(x\), \(y\), and \(z\) terms.

#### For \(y^2 - 2y\):
Complete the square:
\[
y^2 - 2y = (y - 1)^2 - 1
\]

#### For \(x^2 - \frac{50}{3}x\):
Complete the square:
\[
x^2 - \frac{50}{3}x = \left( x - \frac{25}{3} \right)^2 - \frac{625}{9}
\]

#### For \(z^2 + \frac{22}{3}z\):
Complete the square:
\[
z^2 + \frac{22}{3}z = \left( z + \frac{11}{3} \right)^2 - \frac{121}{9}
\]

### Step 5: Rewrite the equation
Substitute the completed squares back into the equation:

\[
(y - 1)^2 - 1 + \left( x - \frac{25}{3} \right)^2 - \frac{625}{9} + \left( z + \frac{11}{3} \right)^2 - \frac{121}{9} + 47 = 0
\]

### Step 6: Combine constants
Now, combine the constants on the right-hand side:

\[
-1 - \frac{625}{9} - \frac{121}{9} + 47 = -1 - \frac{746}{9} + 47 = \frac{-9 - 746 + 423}{9} = \frac{-332}{9}
\]

Thus, the equation becomes:

\[
(y - 1)^2 + \left( x - \frac{25}{3} \right)^2 + \left( z + \frac{11}{3} \right)^2 = \frac{332}{9}
\]

### Final Sphere Equation
This is the equation of a sphere with center \(\left(\frac{25}{3}, 1, -\frac{11}{3}\right)\) and radius \(\sqrt{\frac{332}{9}} = \frac{\sqrt{332}}{3}\).

Would you like more details on completing the square or the geometric interpretation?

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Here are five related questions:
1. How do you complete the square for terms with coefficients?
2. What is the formula for the volume of a sphere?
3. How do you derive the general form of a sphere from its center and radius?
4. What is the geometric interpretation of the center and radius of a sphere?
5. How do you convert between different forms of quadratic equations in 3D?

**Tip:** When simplifying equations, always check if terms can be factored or grouped together to reduce complexity early.

Learn how to simplify the equation y^2 - 10y + x^2 + 2x + z^2 + 35 - 6z = 4y^2 - 16y + 4x^2 + 4z^2 + 16z + 176 - 48x into the standard form of a sphere. This detailed solution covers algebraic manipulation, completing the square, and combining constants. Suitable for students in Grades 10-12.

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