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)2 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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