Squaring a Sum
To find (a + b)2, the square of a sum, we can use FOIL on (a + b)(a + b):
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)2 is not equal to the sum of the squares a2 + b2. The square of a sum has the middle term 2ab.
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.
Squaring a sum
Square each sum, using the new rule.
a) (x + 5)2
b) (2w + 3)2
c) (2y4 + 3)2
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
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.
You can use
to learn a trick for squaring a number that ends in 5. For example, to find 252, find 20 Â· 30 + 25 or 625. More simply, to find 352, find 3 Â· 4 = 12 and follow that by 25: 352 = 1225.