# Completing the Square

Completing the Square

Completing the square of a quadratic function is useful for solving the equation , for plotting the graph of the function, or for finding the center of the circle (or other conic section) that is the graph of .

a+bx+c

2

x

a+bx+c=0

2

x

A+Bxy+C+Dx+Ey+F=0

2

x

2

y

Here is how completing the square works in words:

(1) Factor out from the first two terms, .

a

a+bx

2

x

(2) Add the constant term = in order to be able to factor as a square. Subtract from to keep equality.

2

a

b

2a

2

b

4a

a+bax+

2

x

2

b

2a

2

b

4a

c

(3) Factor to get the result, +c-.

2

ax+

b

2a

2

b

4a

Here it is in symbols:

a+bx+c=a+x+c=a+x++c-=+c-.

2

x

2

x

b

a

2

x

b

a

2

b

2a

2

b

4a

2

ax+

b

2a

2

b

4a