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## Perfect Square – Square Of A Binomial

When the binomial is squared, the result we get is a trinomial. To square a binomial is to multiply the binomial by itself. Consider we have a simple binomial “a + b” and we want to multiply this binomial by itself. To show multiplication, the binomial can be written in the following steps:

(a + b) (a + b) or (a + b)²

The above multiplication can be done using “FOIL” method or best square formula.

**FOIL Method:**

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]

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

**Formula Method:**

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

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

It can be remembered 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 “2*3n * 5” which is “30n” and finally add the square of the second term “5” which is “25”. Writing this all in one step solves the square of the binomial. Let’s all write together;

(3n + 5)² = 9n² + 30n + 25

which is (3n)² + 2 * 3n * 5 + 5²

For example, if there is a negative sign between the terms of a binomial, the second term becomes negative;

(a – b)² = a² – 2ab + b²

The example given will change;

(3n – 5)² = 9n² – 30n + 25

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

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

Examples: (2x + 3y)²

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

= (2x)² + 2 * (2x) * (3y) + (3y)²

= 4x² + 12xy + 9y²

If the sign changes to negative, the process is still the 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 about multiplying a binomial by itself or finding the square of a binomial.

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