Completing the Square
Any quadratic equation can be solved using a technique called completing the square.
To use this method, we will rewrite the quadratic equation so that one side
is a perfect square. Then we solve using the square root property.
Before we solve a quadratic equation in this way, letâ€™s learn how to
complete the square.
To â€œcomplete the squareâ€ means to transform a binomial of the form
x^{2} + bx into a perfect square trinomial by adding a constant term.
For example, letâ€™s complete the square for x^{2} + 6x.
We will use rectangles, called algebra tiles, to visualize the process.
â€¢ A square tile measuring x units on a side has area x^{2}. We will use this
tile to represent x^{2}, the first term of x^{2} + 6x.
â€¢ A rectangular tile that is x units tall and 1 unit wide has an area of 1x.
Since 6 Â· 1x is 6x, we will use six of these tiles to represent the second
term of x^{2} + 6x.
Placed side by side, the tiles form a rectangle that represents x^{2}
+ 6x.
Now, we try to rearrange the tiles to form a square.
To do this, we move onehalf of the tall thin tiles.
However, the result is not a complete square because the lower right
portion is missing.
The missing piece is a 3by3 square.
Thus, to â€œcomplete the square,â€ we must add 3 Â· 3
= 9 unit tiles.
With the 9 new tiles, the area of the entire square is
(x + 3)(x + 3) = (x + 3)^{2} = x^{2} + 6x + 9.
By adding 9 to x^{2} + 6x, we have created the perfect square (x + 3)^{2}.
Letâ€™s review the process we used to
complete the square.
We moved onehalf of the xtiles. 
x^{2} + 6x + ?
Multiply the coefficient of x by

Then we filled in the remaining
space with a square of unit tiles. 
Square the result.
3^{2} = 9 
We added nine tiles to complete the
square. 
x^{2} + 6x + 9
= (x + 3)(x + 3)
= (x + 3)^{2} 
This process holds in general:
To find the number needed to complete the square, multiply the coefficient
of the xterm by
, and then square the result.
