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Square of a Binomial

The square of a binomial is always a trinomial.  It will be helpful to memorize these patterns for writing squares of binomials as trinomials.

( a + b ) 2 = a 2 + 2 a b + b 2

( a b ) 2 = a 2 2 a b + b 2

Examples:

Square each binomial.

a ) ( x + 4 ) 2

( x + 4 ) 2 = x 2 + 2 ( x 4 ) + 4 2 = x 2 + 8 x + 16

b ) ( 2 y 3 ) 2

( 2 y 3 ) 2 = ( 2 y ) 2 2 ( 2 y 3 ) + 3 2 = ( 2 y ) 2 2 ( 6 y ) + 3 2 = 4 y 2 12 y + 9

c ) ( 3 p 2 q 2 ) 2

( 3 p 2 q 2 ) 2 = ( 3 p ) 2 2 ( 3 p 2 q 2 ) + ( 2 q 2 ) 2 = 9 p 2 2 ( 6 p q 2 ) + 4 q 4 = 9 p 2 12 p q 2 + 4 q 4

If the coefficients of a trinomial a x 2 + b x + c satisfy the equation

c = ( b 2 a ) 2 ,

then the trinomial is the perfect square of the binomial

a x + b 2 .

Example 1:

Factor, if possible.

x 2 14 x + 49

Here, a = 1 , b = 14 , and c = 49 . We have:

( b 2 a ) 2 = ( 14 2 1 ) 2 = ( 7 ) 2 = 49 = c

So, the trinomial is a perfect square:

x 2 14 x + 49 = ( x 7 ) 2

You can verify this using FOIL .

Example 2:

Factor, if possible.

9 w 4 + 12 w 2 + 4

Here, a = 9 , b = 12 , and c = 4 . (We can treat w 2 as x , and not worry about the fourth power.)

( b 2 a ) 2 = ( 12 2 9 ) 2 = ( 12 6 ) 2 = 4 = c

So, the trinomial is a perfect square:

9 w 4 + 12 w 2 + 4 = ( 3 w 2 + 2 ) 2

This can also be verified using FOIL.