How do you find the vertex of #(x + 6)^2 = -36(y − 3)#?

1 Answer
Jan 30, 2016

vertex#=(-6,3)#

Explanation:

1. Expand both sides of the equation.

#(x+6)^2=-36(y-3)#

#x^2+12x+36=-36y+108#

2. Isolate for y.
Recall that general equation for a quadratic equation in standard form is #y=ax^2+bx+c#. Thus, isolate for #y#.

#x^2+12x+36-108=-36y#

#x^2+12x-72=-36y#

#y=-1/36x^2-1/3x+2#

3. Factor -1/36 from the first two terms.
To find the vertex, we must complete the square. We can do this by first factoring #-1/36# from the first two terms.

#y=-1/36(x^2+12x)+2#

4. Rewrite the bracketed terms as a perfect square trinomial.
The value of #c# in a perfect square trinomial is #(b/2)^2#. Thus, divide #12# by #2# and square the value.

#y=-1/36(x^2+12x+((12)/2)^2)+2#

#y=-1/36(x^2+12x+36)+2#

5. Subtract 36 from the perfect square trinomial.
We cannot just add #36# to the perfect square trinomial, so we must subtract #36# from the #36# we just added.

#y=-1/36(x^2+12x+36# #color(red)(-36))+2#

6. Multiply -36 by -1/36 to move -36 out of the brackets.

#y=-1/36(x^2+12+36)+2(-36)*(-1/36)#

7. Simplify.

#y=-1/36(x^2+12+36)+2[(-color(red)cancelcolor(black)36)*(-1/color(red)cancelcolor(black)36)]#

#y=-1/36(x^2+12+36)+2+1#

#y=-1/36(x^2+12+36)+3#

8. Factor the perfect square trinomial.
The final step to finding the vertex is to factor the perfect square trinomial. This will tell you the #x# coordinate of the vertex. The #y# coordinate of the vertex, #3#, has already been found.

#y=-1/36(x+6)^2+3#

#:.#, the vertex is #(-6,3)#.