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Perfect Square – Square Of A Binomial
When a binomial is squared, the result obtained is a trinomial. Squaring a binomial means multiplying the binomial by itself. Consider that we have the simplest binomial “a + b” and we want to multiply this binomial by itself. To show multiplication, the binomial can be written as in the step below:
(a + b) (a + b) or (a + b)²
The above multiplication can be done using the “FOIL” method or using the perfect square formula.
The FOIL method:
Let’s simplify the multiplication above 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 for solving the square of a binomial.
The formula method:
By the method of the formula, the final result of the 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.
Remember that (a + b)² = a² + 2ab + b²
It can be memorized as;
(first term)² + 2 * (first term) * (second term) + (second term)²
Consider that 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 the second term “5” which is “25 “. Writing it all down in one step solves the binomial square. Let’s write everything together;
(3n + 5)² = 9n² + 30n + 25
Let (3n)² + 2 * 3n * 5 + 5²
For example if there is a negative sign between the terms of the binomial then the second term becomes the negative as;
(a – b)² = a² – 2ab + b²
The example given will become;
(3n – 5)² = 9n² – 30n + 25
Again, remember the following to find the square of a binomial directly by the formula;
(first term)² + 2 * (first term) (second term) + (second term)²
Examples: (2x + 3a)²
Solution: The first term is “2x” and the second term is “3y”. Let’s follow the formula to perform the square of the given binomial;
= (2x)² + 2 * (2x) * (3a) + (3a)²
= 4x² + 12xy + 9y²
If the sign is changed to negative, the procedure 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
It involves multiplying a binomial by itself or finding the square of a binomial.
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