Question

vertex of the parabola 9 x square - 6 X + 36 Y + 9 equal to zero is

Answer 1

9 x² - 6 x + 36 y + 9 = 0

(3x-1)² = -4×9 (y+2/9)

hence vertex = (1/3, -2/9)

Answer 2

The vertex of the parabola is (1/3, -2/9)

Gvien equation,

              $9x^{2} -6x + 36y + 9 = 0\\9x^{2} -6x =  -36y - 9\\x^{2} - \frac{2}{3}x = \frac{-36}{9}y - \frac{2}{9}\\  x^{2} - \frac{2}{3}x = -4y-1\\Adding (\frac{1}{3}) ^{2} on both sides\\x^{2} - \frac{2}{3}x +  (\frac{1}{3}) ^{2}  = -4y-1 +  (\frac{1}{3}) ^{2} \\(x-\frac{1}{3}) ^{2} = -4y - 1 +  (\frac{1}{3}) ^{2} \\(x-\frac{1}{3}) ^{2} = -4y - \frac{8}{9} \\(x-\frac{1}{3}) ^{2} = -4(y + \frac{2}{9})\\It is in the form of \\(x-h}) ^{2} = -4a(y -k)$

Here,

h = 1/3, k = -2/9

Vertex = (h, k) = (1/3, -2/9)