What is the vertex of y= 3x^2-2x-(3x+2)^2?

1 Answer
Dec 17, 2015

The vertex is at (x,y)=(-7/6,25/6)

Explanation:

Probably the easiest way to do this is to convert the given equation into "vertex form:
color(white)("XXX")y=color(orange)(m)(x-color(red)(a))^2+color(blue)(b) with vertex at (color(red)(a),color(blue)(b))

Given:
color(white)("XXX")y=3x^2-2x-(3x+2)^2

Expand and simplify the expression on the right side:
color(white)("XXX")y=3x^2-2x-(9x^2+12x+4)

color(white)("XXX")y= -6x^2-14x-4

Extract the m factor
color(white)("XXX")y=color(orange)((-6))(x^2+14/6x)-4

Complete the square
color(white)("XXX")y=color(orange)((-6))(x^2+14/6x+14^2/12^2) - 4 +6*(14^2/(12^2))

color(white)("XXX")y=color(orange)((-6))(x+color(red)(14/12))^2 -4 +196/24

color(white)("XXX")y=color(orange)((-6))(x-color(red)((-7/6)))+color(blue)(25/6)
graph{3x^2-2x-(3x+2)^2 [-3.342, 2.815, 2.025, 5.102]}