Reduce the equation to standard form, classify the surface x^2-y^2+z^2-4x-2z=0

I need step by step to the answer ((x-2)^2)/5 -(y^2)/5 +((z-1)^2)/5 =1

it's just the good old complete-the-square method from Algebra 2, right?

x^2-y^2+z^2-4x-2z=0
x^2-4x - y^2 + z^2-2z = 0
x^2-4x+4 - y^2 + z^2-2z+1 = 4+1
(x-2)^2 - y^2 + (z-1)^2 = 5
Think you can manage that last step?

To reduce the equation to standard form and classify the surface, follow these steps:

Step 1: Rearrange the equation by grouping the variables:
x^2 - 4x - y^2 + z^2 - 2z = 0

Step 2: Complete the square for the x terms:
(x^2 - 4x) + (y^2 - z^2) + (z^2 - 2z) = 0

Step 3: To complete the square for the x terms, take half of the coefficient of x (-4/2 = -2) and square it (4):
(x^2 - 4x + 4) + (y^2 - z^2) + (z^2 - 2z) = 4

Step 4: Repeat the process to complete the square for the z terms:
(x^2 - 4x + 4) + (y^2 - z^2) + (z^2 - 2z + 1) = 4 + 1

Step 5: Simplify and combine like terms:
(x - 2)^2 + (y^2 - z^2) + (z - 1)^2 = 5

Step 6: Divide the equation by 5:
((x - 2)^2)/5 + (y^2 - z^2)/5 + ((z - 1)^2)/5 = 1

Now the equation is in standard form: ((x - 2)^2)/5 + (y^2 - z^2)/5 + ((z - 1)^2)/5 = 1

To classify the surface, observe the coefficients of the squared terms:
- The coefficient of (x - 2)^2 is positive, so the surface is an ellipsoid in the x-direction.
- The coefficient of (y^2 - z^2) is 0, so the surface is perpendicular to the y-z plane.
- The coefficient of (z - 1)^2 is positive, so the surface is an ellipsoid in the z-direction.

Therefore, the surface is an elliptical cylinder oriented along the x-axis.

To reduce the given equation to standard form and classify the surface, we need to complete the square for each variable (x, y, and z). Here's a step-by-step explanation:

1. Start with the given equation: x^2 - y^2 + z^2 - 4x - 2z = 0.

2. Group the x and z terms together, and complete the square for each variable:

(x^2 - 4x) - (y^2) + (z^2 - 2z) = 0.

To complete the square for x, take half the coefficient of x (-4) and square it: (-4/2)^2 = 4.

To complete the square for z, take half the coefficient of z (-2) and square it: (-2/2)^2 = 1.

Add these values inside parentheses, and add them outside the parentheses to maintain balance:

(x^2 - 4x + 4) - (y^2) + (z^2 - 2z + 1) = 4 - 1.

Simplify the right side of the equation:

(x^2 - 4x + 4) - (y^2) + (z^2 - 2z + 1) = 3.

3. Factor the perfect square trinomials:

(x - 2)^2 - (y^2) + (z - 1)^2 = 3.

4. To put the equation in standard form, divide each term by the constant on the right side of the equation, which is 3:

(x - 2)^2/3 - (y^2)/3 + (z - 1)^2/3 = 1.

5. Comparing this result with the standard equation for a surface (x^2/a^2 + y^2/b^2 + z^2/c^2 = 1), we can determine the classification of the surface:

The given equation represents an ellipsoid with semi-axes lengths of sqrt(3) in the x-direction, sqrt(3) in the y-direction, and sqrt(3) in the z-direction.