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WOLFRAM|DEMONSTRATIONS PROJECT

Completing the Square

​
a
-3
b
-3
c
-3
-3
2
x
-3x -3 =
-3(
2
x
+x) -3 =
-3
2
x
+x+
1
4
-
9
4
=
-3
2
x+
1
2
-
9
4
The vertex is at -
1
2
​, -
9
4
The complex roots are x = -
1
2
±
3
2
≈ -0.500 ± 0.866 .
Completing the square of a quadratic function
a
2
x
+bx+c
is useful for solving the equation
a
2
x
+bx+c=0
, 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
2
x
+Bxy+C
2
y
+Dx+Ey+F=0
.
Here is how completing the square works in words:
(1) Factor out
a
from the first two terms,
a
2
x
+bx
.
(2) Add the constant term
2
a
b
2a
=
2
b
4a
in order to be able to factor
a
2
x
+bax+
2
b
2a
as a square. Subtract
2
b
4a
from
c
to keep equality.
(3) Factor to get the result,
2
ax+
b
2a
+c-
2
b
4a
.
Here it is in symbols:
a
2
x
+bx+c=​​a
2
x
+
b
a
x+c=​​a
2
x
+
b
a
x+
2
b
2a
+c-
2
b
4a
=​​
2
ax+
b
2a
+c-
2
b
4a
.
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