
Identify the surface defined by the following equation. x^2 + y^2 + 4z^2 +12x = -35

The given equation is:

\[
x^2 + y^2 + 4z^2 + 12x = -35
\]

### Step 1: Complete the square for the \(x\)-term.

The quadratic expression in \(x\) is \(x^2 + 12x\). To complete the square:

1. Take half of the coefficient of \(x\), which is \(12\), and divide by 2: \(12/2 = 6\).
2. Square this result: \(6^2 = 36\).

Now, we rewrite the \(x^2 + 12x\) as:
\[
x^2 + 12x = (x + 6)^2 - 36
\]

So the equation becomes:
\[
(x + 6)^2 - 36 + y^2 + 4z^2 = -35
\]

### Step 2: Simplify the equation.

Add 36 to both sides:
\[
(x + 6)^2 + y^2 + 4z^2 = 1
\]

### Step 3: Identify the surface.

This equation is of the form:
\[
\frac{(x + 6)^2}{1} + \frac{y^2}{1} + \frac{z^2}{\frac{1}{4}} = 1
\]

This is the equation of an **ellipsoid**. The ellipsoid is centered at \((-6, 0, 0)\) and has semi-axes lengths of 1 in the \(x\)- and \(y\)-directions, and \(1/2\) in the \(z\)-direction.

### Summary:

The surface is an **ellipsoid** with center at \((-6, 0, 0)\), with semi-axes lengths 1 in the \(x\)- and \(y\)-directions, and \(1/2\) in the \(z\)-direction.

Would you like more details or have any further questions?

Here are five related questions to deepen your understanding:
1. How would the equation change if the ellipsoid were centered at the origin?
2. What are the semi-axes lengths of the ellipsoid in general form?
3. How does the equation of an ellipsoid differ from that of a hyperboloid?
4. Can you transform this ellipsoid equation into cylindrical coordinates?
5. What is the geometric significance of completing the square in quadratic equations?

**Tip:** Completing the square is a useful technique for transforming quadratic equations into standard forms, making it easier to identify geometric properties.

Identify the surface defined by the following equation. x^2 + y^2 + 4z^2 + 12x = -35

This detailed solution explores the equation x^2 + y^2 + 4z^2 + 12x = -35, teaching how to identify the surface as an ellipsoid using the completing the square method. The ellipsoid has a center at (-6, 0, 0) and semi-axes of 1 and 1/2. Ideal for high school students studying Algebra and Geometry.

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