## jedai17 2 years ago can someone answer this quadric surface using completing the square ?thank you :) x^2 + 4y^2 - z^2 - 6x + 8y + 4z = 0

1. amistre64

get your like variables together and complete the squares .... which part is giving you pause?

2. amistre64

you can pull it apart into 3 quads if you like .... that might be less data to cull thru.

3. jedai17

i don't know how to solve this..i'm sorry

4. amistre64

do you know how to complete a square?

5. jedai17

it means that it will have a two factors in each variables..all in all it has a 6 factors?

6. jedai17

it is x^2 - 6x , 4y^2 + 8y and z^2 + 4z?

7. amistre64

lets go ahead and split this up into 3 different parts: x^2 + 4y^2 - z^2 - 6x + 8y + 4z = 0 x^2 -6x 4y^2 +8y -z^2 + 4z now that its organized better, you just work each one in its own right .... complete the square for each part

8. amistre64

the key is, do you know how to complete a square? :)

9. jedai17

ah.. its -z^2..

10. jedai17

i'm sorry i forgot how to complete a square..

11. amistre64

basically, you add and subtract a useful form of zero. square half the "x" term ..... for example: x^2 -6x ; the x term is -6, -6/2 = -3, (-3)^2 = 9. Lets add and subtract 9 x^2 -6x + 9 - 9; and condense the square (x-3)^2 - 9

12. amistre64

z will be trick only in that you have to account for the -

13. jedai17

x^2 -6x ; the x term is -6, how it become -6/2?

14. amistre64

we are trying to make a complete square, a perfect square. numbers of the form (a+b)^2 are perfect squares: (2+5)^2 = 49, (3+1)^2 = 16, ... we want to condense this thing back into some (a + b)^2 form, thats all. In order to do that we have to modify its form, but not its value. Since adding zero changes nothing ... its useful to us. BUT what form of zero? lets expand (a+b)^2 = a^2 +2ab + b^2 ; let a=x (x+b)^2 = x^2 +2bx + b^2 notice that the b^2 is missing above. and we know that the "x" term in the middle is equal to 2b. 2b = -6 b = -6/2 = -3 b^2 = 9

15. amistre64

so, adding and subtract 9; 9-9 = 0 right? will allow us to "complete" the square and alter its form ... not its value

16. amistre64

4y^2 +8y 2b = 8 b = 8/2 = 4 b^2 = 16, +16-16 = 0 4y^2 +8y + (16-16) therefore 4y^2 +8y = (2y +4)^2 - 16

17. jedai17

oh..i see. thanks i will solve this..

18. jedai17

btw do you know about contour plotting?

19. amistre64

i would need a little more information to see if i have contour plotting rolling around in my head someplace :)

20. jedai17

do you know what kind of contour plot is this if f(x,y) = $\sqrt{x^2 + y^2}$ ?

21. amistre64

so the height, or z, is dependent on x and y inputs if we hold one input constant, we can generate a cross section plot and such, are some ideas ive got z = sqrt(x^2+y^2) z^2 = x^2 + y^2 is a circle, and so im gonna extrapolate that this is a sphere or hemisphere as a guess

22. amistre64

http://web.monroecc.edu/manila/webfiles/calcnsf/javacode/calcplot3d.htm might be useful :)

23. amistre64

well it use to work fine, i guess they changed some stuff about

24. amistre64

http://www.wolframalpha.com/input/?i=sqrt%28x%5E2%2By%5E2%29 hmmm, its conic not spherical. it comes to a point at the origin and makes bigger circles as z gets bigger and bigger

25. jedai17

our topic to this is about contour plotting..

26. amistre64

|dw:1361892976311:dw|

27. amistre64

im not up to date on what contour plotting is, this is mostly from an old, decripid memory :)

28. amistre64

for your private message z = k z = x^2 + y for what values of x^2 + y does k = the stated set of values

29. amistre64

x^2 + y = -2 when y=-x^2 -2|dw:1361893200001:dw|

30. amistre64

in other words, the contour follows its reflection in the xy plane if that makes sense

31. amistre64

for any value of k, the reflected graph in the xy plane changes its y intercept so it will float up and down the y axis as needed

32. jedai17

what kind of contour plot is that?

33. amistre64

it looks like a repeition of a parabola to me, so its a type of cylindar right?

34. amistre64

|dw:1361894575970:dw|

35. jedai17

i don't know..but i heard that there is a graph named paraboloid, elliptical paraboloid etc..

36. jedai17

but i think it is elliptical paraboliod because you said that it is cylindar..