When a binomial is squared, the result we get is a trinomial. Squaring a binomial means, multiplying the binomial by itself. Consider we have a simplest binomial “a + b” and we want to multiply this binomial by itself. To show the multiplication the binomial can be written as in the step below:

(a + b) (a +b) or (a + b)²

The above multiplication can be carried out using the “FOIL” method or using the perfect square formula.

The FOIL method:

Let’s simplify the above multiplication using the FOIL method as explained below:

(a + b) (a +b)

= a² + ab + ba + b²

= a² + ab + ab + b² [Notice that ab = ba]

= a² + 2ab + b² [As ab + ab = 2ab]

That is the “FOIL” method to solve the square of a binomial.

The Formula Method:

By the formula method the final result of the multiplication for (a + b) (a + b) is memorized directly and applied it to the similar problems. Let’s explore the formula method to find the square of a binomial.

Commit to memory that (a + b)² = a² + 2ab + b²

It can be memorized as;

(first term)² + 2 * (first term) * (second term) + (second term)²

Consider we have the binomial (3n + 5)²

To get the answer, square the first term “3n” which is “9n²”, then add the “2* 3n * 5” which is “30n” and finally add the square of second term “5” which is “25”. Writing all this in a step solves the square of the binomial. Let’s write it all together;

(3n + 5)² = 9n² + 30n + 25

Which is (3n)² + 2 * 3n * 5 + 5²

For example if there is negative sign between he terms of the binomial then the second term becomes the negative as;

(a – b)² = a² – 2ab + b²

The given example will change to;

(3n – 5)² = 9n² – 30n + 25

Again, remember the following to find square of a binomial directly by the formula;

(first term)² + 2 * (first term) (second term) + (second term)²

Examples: (2x + 3y)²

Solution: First term is “2x” and the second term is “3y”. Let’s follow the formula to carried out the square of the given binomial;

= (2x)² + 2 * (2x) * (3y) + (3y)²

= 4x² + 12xy + 9y²

If the sign is changed to negative, the procedure is still same but change the central sign to negative as shown below:

(2x – 3y)²

= (2x)² + 2 * (2x) * (- 3y) + (-3y)²

= 4x² – 12xy + 9y²

That is all about multiplying a binomial by itself or to find the square of a binomial.

Source by Manjit Singh Atwal