What is Completing the Square? Definition and Examples

What is "completing the square"? Completing the square is the process of changing an expression like x2 + bx into a perfect square trinomial by adding (b/2)2 to x2 + bx.

For example, you can change x2 + 4x into a perfect square trinomial by adding (4/2)2 or 22 = 4 to x2 + 4x.

x2 + 4x + 4 = (x + 2)(x + 2) = (x + 2)2, so x2 + 4x + 4 is a perfect square trinomial.

Completing the square using algebra tiles

Algebra tiles

Using the algebra tiles above, complete the square for x2 + 4x.

First, use the algebra tiles to model x2 + 4x.

x squared plus four x

Completing the square with algebra tiles means that you will rearrange the tiles and add more tiles so that everything will look like a square. 

a. Rearrange the tiles above so that it begins to look like a square.

So far, you can see that it is beginning to look like a square. However, it is not a square just yet.

b. You can complete the square by adding 4 of the small green squares where you see the empty space.

Completing the square

Completing the square with a couple more examples 

1. For x2 + 6x + n, find the value of n that will complete the square.

n = (6/2)2

n = (3)2

n = 9

2. For x2 + -8x + n, find the value of n that will complete the square.

n = (-8/2)2

n = (-4)2

n = 16

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