The Square of a Binomial
To compute (a + b)^{2}, the square of a binomial, we can write it as
(a + b)(a + b) and use FOIL:
(a + b)^{2} 
= (a + b)(a + b) 

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

= a^{2} + 2ab + b^{2} 
So to square a + b, 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}).
The square of a binomial occurs so frequently that it is helpful to learn this
new rule to find it. The rule for squaring a sum is given symbolically as
follows.
The Square of a Sum
(a + b)^{2 }= a^{2} + 2ab + b^{2}
Example 1
Using the rule for squaring a sum
Find the square of each sum.
a) (x + 3)^{2}
b) (2a + 5)^{2}
Solution
a) (x + 3)^{2 }= 
x^{2} 
+ 2(x)(3) 
+ 3^{2} 
= x^{2} + 6x + 9 

↑ 
↑ 
↑ 


Square of first 
Twice the procuct 
Square of last 

b) (2a + 5)^{2} 
= (2a)^{2} + 2(2a)(5) + 5^{2} 

= 4a^{2} + 20a + 25 
Caution
Do not forget the middle term when squaring a sum. The equation (x + 3)^{2}
= x^{2} + 6x + 9 is an identity, but (x + 3)^{2} = x^{2}
+ 9 is not an identity. For example, if x = 1 in (x + 3)^{2} = x^{2}
+ 9, then we get 4^{2} = 1^{2} + 9, which is false.
When we use FOIL to find (a  b)^{2}, we see that
(a  b)^{2} 
= (a  b)(a  b) 

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

= a^{2}  2ab + b^{2} 
So to square a  b, we square the first term (a^{2}), subtract twice
the product of the two terms (2ab), and add the square of the last term (b^{2}).
The rule for squaring a difference is given symbolically as follows.
The Square of a Difference
(a  b)^{2 }= a^{2}  2ab + b^{2}
Example 2
Using the rule for squaring a difference
Find the square of each difference.
a) (x  4)^{2}
b) (4b  5y)^{2}
Solution
a) (x  4)^{2} 
= x^{2}  2(x)(4) + 4^{2} 

= x^{2}  8x + 16 
b) (4b  5y)^{2} 
= (4b)^{2}  2(4b)(5y) + (5y)^{2} 

= 16b^{2}  40by + 25y^{2} 