Completing the Square with Coefficient on X^2

[fuzzface89]

Homework Statement

Consider all possible surfaces that can be formed from the variations of

\[PlusMinus]x^2\[PlusMinus]4y^2\[PlusMinus]z^2+2x+8y+6z==-6,

that is, find and describe all the different quadric surfaces you can make from this equation by using different signs on the degree 2 terms. For each one, find the standard form of the equation.

I am having problems completing the square when the coefficient is negative.

The Attempt at a Solution

For the case where the y-coefficient is negative:

(x^2 + 2x + 1) + (-4y^2 + 8y + _ ) + (z^2 +6z + 9) = -6 + 1 + _ + 9

dividing by -4 gives: (y^2 - 2y + _ )

Adding (b/2a)^2: (y^2 - 2y + 1)

Condensing: (y - 1)^2

Now, does the coefficient 4 come back as positive or negative, as in
4(y - 1)^2 or -4(y - 1)^2

Also, when adding to the other side of the equation, does the -4 multiply to the added quantity?

Thanks for any help

 

[Mark44]

(-4y² + 8y )
= -(4y² - 8y + __) + something

You're going to add a positive number inside the parentheses, but you have really added a negative number, so to keep the expression in balance, add the positive of that negative number.

Make sense?

 

[Mark44]

What else you should do to make your life simpler is to bring out the coefficient on the 2nd degree term so that you have
-4(y² - 2y + __) + something

Just keep track of what you really added to the overall expression.

 

[fuzzface89]

Okay, thanks for helpin me out