Complete the square and reduce to the following form: y-b=A(x-a)^2 or x-a=A(y-b)^2

Question

9x²-6y-9-x=0

Answer 1

9x^2 - 6y - 9 - x = 0

solve for y, add 6y to both sides

9x^2 -9 -x = 6y, switch the sides

6y = 9x^2 -x -9

divide by 6

y = 9x^2/6 - x/6 - 9/6

factor out 9/6 from the x^2 and x terms

y = (9/6)(x^2 -(1/9)x) -9/6 9/6=3/2

y =(3/2)(x^2 - x/9) -3/2

to complete the square take half of the x term's coefficient, square it, then add it to the expression inside the parentheses, and multiply it by (3/2) and subtract that outside the parentheses. That's the same as adding and subtracting the same number, leaving the right side the same value. Half of 1/9 is 1/18, square it that's 1/18^2.

add it inside the parentheses, that's the same as adding (3/2)(1/18^2) so subtract that same amount outside the parentheses.

y = (3/2)(x^2 - x/9 + 1/18^2) - 3/2 - (3/2)(1/18^2) move the constant terms to the left side and simplify

that's the same as adding 3/2 + (3/2)(1/18^2) to both sides. 3/2(18)^2 = 1/216

y+ 3/2 + 1/216= (3/2)(x- 1/18)^2 3/2+1/216 = 324/216 + 1/216 = 325/216

y + 325/216 = (3/2)(x-1/18)^2

This is an upward opening parabola with vertex (1/18, - 325/216)

b = -325/216 = the parabola opens up because A=3/2>0

b= the minimum point

for y-b = A(x-a)^2 b=--325/216, A = 3/2 a= 1/18

y- (-325/216) = (3/2)(x-1/18)^2

the vertex is the minimum point. It's upward opening

Answer 2

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