1
[tex] 3x^2 + 5y^2 - 12x + 30y + 42 = 0 [/tex]
[tex] 3x^2 - 12x + 5y^2 + 30y = - 42 [/tex]
[tex] 3(x^2 - 4x) + 5(y^2 + 6y) = -42 [/tex]
[tex] 3(x^2 -4x + 4) + 5(y^2 + 6y + 9) = -42 + 12 + 45 [/tex]
[tex] 3(x-2)^2 + 5(y+3)^2 = 15 [/tex]
[tex] \dfrac{(x-2)^2}{5} + \dfrac{(y+3)^2}{3} = 1 [/tex]
Ellipse, same signs different coefficients on [tex] x^2 [/tex] and [tex] y^2 [/tex]
2
[tex] 9x^2 -36x - 4y^2 + 8y = 4 [/tex]
[tex] 9(x^2 - 4x + 4) - 4(y^2 - 2y + 1) = 4 + 36 - 4 [/tex]
[tex] 9(x- 2)^2 - 4(y - 1^2) = 36 [/tex]
Hyperbola, opposite signs on [tex] x^2 [/tex] and [tex] y^2 [/tex]
3
[tex] y = x^2 +2x + 3 = (x + 1)^2 + 2 [/tex]
parabola
4
[tex] x^2 - 4 x + y^2 + 4y = 4 [/tex]
[tex] (x - 2)^2 + (y+2)^2 = 4 + 4 + 4 = 12 [/tex]
equal coefficents on [tex] x^2 [/tex] and [tex] y^2 [/tex], circle
5
[tex] -9x^2 - 18x + 4y^2 - 8y = 41 [/tex]
[tex] -9(x^2 -2x + 1) + 4(y^2 - 2y + 1) = 41 - 9 + 4 [/tex]
[tex] -9(x-1)^2 + 4(y-1)^2 = 36 [/tex]
hyperbola, opposite signs
6
[tex] -4x = y^2 - 2y - 11 = (y-1)^2 - 12 [/tex]
That's a parabola, sideways open to the left
7
[tex] 2(x^2 + 6x) + 3(y^2 - 8y) = -60 [/tex]
[tex] 2(x^2 + 6x + 9) + 3(y^2 - 8y + 16) = -60 + 18 + 48 [/tex]
[tex] 2(x+3)^2 + 3(y - 4)^2 = 6 [/tex]
[tex] \dfrac{(x+3)^2}{3} + \dfrac{(y-4)^2}{2} = 1 [/tex]
Ellipse, different coefficients
8
[tex] 16x^2 - 32x - y^2 - 6y = 57 [/tex]
[tex] 16(x^2 - 2x + 1) - (y^2 + 6y + 9) = 57 + 16 - 9 = 64 [/tex]
opposite signs, hyperbola
You'll have to do the graphing yourself, sorry
