# 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 .

ax+bx+c

2

ax+bx+c=0

2

Ax+Bxy+Cy+Dx+Ey+F=0

2

2

Here is how completing the square works in words:

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

a

ax+bx

2

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

a=

b

2a

2

b

2

4a

ax+b/ax+

2

b

2a

2

b

2

4a

c

(3) Factor to get the result, .

ax++c-

b

2a

2

b

2

4a

Here it is in symbols:

ax+bx+c=

2

ax+x+c=

2

b

a

ax+x++c-=

2

b

a

b

2a

2

b

2

4a

ax++c-.

b

2a

2

b

2

4a