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The Square of a Sum

To find (a + b)2, the square of a sum, we can use FOIL on (a + b)(a + b):

(a + b)(a + b) = a2 + ab + ab + b2
  = a2 + 2ab + b2

You can use the result a2 + 2ab + b2 that we obtained from FOIL to quickly find the square of any sum. To square a sum, we square the first term (a2), add twice the product of the two terms (2ab), then add the square of the last term (b2).

 

Rule for the Square of a Sum

(a + b)2 = a2 + 2ab + b2

In general, the square of a sum (a + b)is not equal to the sum of the squares a2 + b2. The square of a sum has the middle term 2ab.

 

Helpful hint

To visualize the square of a sum, draw a square with sides of length a + b as shown.

The area of the large square is (a + b)2. It comes from four terms as stated in the rule for the square of a sum.

 

Example 2

Squaring a sum

Square each sum, using the new rule.

a) (x + 5)2

b) (2w + 3)2

c) (2y4 + 3)2

Solution

a) (x + 5)2 = x2 + 2(x)(5) + 52 = x2 + 10x + 25
 

Square of first

Twice the product

Square of last

 

b) (2w + 3)2 = (2w)2 + 2(2w)(3) + 32 = 4w2 + 12w + 9

c) (2y4 + 3)2 = (2y4)2 + 2(2y4)(3) + 32 = 4y8 + 12y4 + 9

Caution

Squaring x + 5 correctly, as in Example 2(a), gives us the identity (x + 5)2 = x2 + 10x + 25, which is satisfied by any x. If you forget the middle term and write (x + 5)2 = x2 + 25, then you have an equation that is satisfied only if x = 0.

Helpful hint

You can use

(x + 5)2 = x2 + 10x + 25
  = x(x + 10) + 25

to learn a trick for squaring a number that ends in 5. For example, to find 252, find 20 · 30 + 25 or 625. More simply, to find 352, find 3 · 4 = 12 and follow that by 25: 352 = 1225.

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