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) 
= a^{2} + ab + ab + b^{2} 

= a^{2} + 2ab + b^{2} 
You can use the result a^{2} + 2ab + b^{2} that we obtained from FOIL to quickly find
the square of any sum. To square a sum, we square the first term (a^{2}), add twice the
product of the two terms (2ab), then add the square of the last term (b^{2}).
Rule for the Square of a Sum
(a + b)^{2} = a^{2} + 2ab + b^{2}
In general, the square of a sum (a + b)^{2 }is not equal to the sum of the squares
a^{2} + b^{2}. 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) (2y^{4} + 3)^{2}
Solution
a) (x + 5)^{2} 
= x^{2} + 
2(x)(5) 
+ 5^{2} 
= x^{2} + 10x + 25 

↑
Square of first 
↑
Twice the product 
↑
Square of last 

b) (2w + 3)^{2} = (2w)^{2} + 2(2w)(3) + 3^{2} = 4w^{2}
+ 12w + 9
c) (2y^{4} + 3)^{2} = (2y^{4})^{2} + 2(2y^{4})(3)
+ 3^{2} = 4y^{8} + 12y^{4} + 9
Caution
Squaring x + 5 correctly, as in Example 2(a), gives us the
identity
(x + 5)^{2} = x^{2} + 10x + 25, which is satisfied by any x. If you forget the middle term and write
(x + 5)^{2} = x^{2} + 25, then you have an equation that is satisfied only if x
= 0.
Helpful hint
You can use
(x + 5)^{2} 
= x^{2} + 10x + 25 

= x(x + 10) + 25 
to learn a trick for squaring a
number that ends in 5. For
example, to find 25^{2}, find
20 Â· 30 + 25 or 625. More
simply, to find 35^{2}, find
3 Â· 4 = 12 and follow that by 25: 35^{2} = 1225.
