Perfect Square - Square Of A Binomial

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.