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 +2ab+ b 2

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

Examples:

Square each binomial.

a) (x+4) 2

( x+4 ) 2 = x 2 +2( x4 )+ 4 2 = x 2 +8x+16

b) (2y3) 2

( 2y3 ) 2 = ( 2y ) 2 2( 2y3 )+ 3 2 = ( 2y ) 2 2( 6y )+ 3 2 =4 y 2 12y+9

c) ( 3p2 q 2 ) 2

( 3p2 q 2 ) 2 = ( 3p ) 2 2( 3p2 q 2 )+ ( 2 q 2 ) 2 =9 p 2 2( 6p q 2 )+4 q 4 =9 p 2 12p q 2 +4 q 4

If the coefficients of a trinomial a x 2 +bx+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 14x+49

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

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

So, the trinomial is a perfect square:

x 2 14x+49= (x7) 2

You can verify this using FOIL.

Example 2:

Factor, if possible.

9 w 4 +12 w 2 +4

Here, a=9,b=12,andc=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.