The general vertex form of a quadratic equation is y=a(x−h)2+k,
where the vertex of the graph of the function is the point (h,k). To convert this quadratic equation from standard form to vertex form, we will follow the process of completing the square. We will first isolate the x-terms on the right side of the equation, and then factor out the coefficient of the x2-term.
y=12x2+12x−8
y+8=12x2+12x−8+8
y+8=12x2+12x
y+8=12(x2+24x)
The equation is now in the form necessary to complete the square. We will find the square of half the coefficient of the x-term to complete the square.
(242)2=122=144
So, we will use 144 to complete the square.
y+8+72=12(x2+24x+144)
Note that we added 72 to the left side of the equation because on the right side, the 144 is inside parenthesis to be multiplied by 12, and 12(144)=72. Now factor the right side of the equation.
y+80=12(x+12)2
Now, isolate the y-term on the left side of the equation to have the equation in vertex form.
y+80−80=12(x+12)2−80
y=12(x+12)2−80
