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