The (a – b)2 formula says (a – b)2 = a2 – 2ab + b2. It is used to find the square of a binomial. This a minus b whole square formula is one of the commonly used algebraic identities. This formula is also known as the formula for the square of the difference between two terms.
The (a – b)2 formula is used to factorize some special types of trinomials. Let us learn more about a minus b Whole Square along with solved examples in the following section.
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What is (a – b)^2 Formula?
The (a – b)2 formula is also widely known as the square of the difference between the two terms. It says (a – b)2 = a2 – 2ab + b2. This formula is sometimes used to factorize the binomial. To find the formula of (a – b)2, we will just multiply (a – b) (a – b).
(a – b)2 = (a – b)(a – b)
= a2 – ab – ba + b2
= a2 – 2ab + b2
Therefore, (a – b)2 formula is:
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(a – b)2 = a2 – 2ab + b2
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Proof of A minus B Whole Square Formula
Let us consider (a – b)2 as the area of a square with length (a – b). In the above figure, the biggest square is shown with area a2.
To prove that (a – b)2 = a2 – 2ab + b2, consider reducing the length of all sides by factor b, and it forms a new square of side length a – b. In the figure above, (a – b)2 is shown by the blue area. Now subtract the vertical and horizontal strips that have the area a × b. Removing a × b twice will also remove the overlapping square at the bottom right corner twice hence add b2. On rearranging the data we have (a − b)2 = a2 − ab − ab + b2. Hence this proves the algebraic identity (a − b)2 = a2 − 2ab + b2.
Examples on (a – b)^2 Formula
Example 1: Find the value of (x – 2y)2 by using the (a – b)2 formula.
Solution:
To find: The value of (x – 2y)2. Let us assume that a = x and b = 2y. We will substitute these values in (a – b)2 formula: (a – b)2 = a2 – 2ab + b2 (x – 2y)2 = (x)2 – 2(x)(2y) + (2y)2 = x2 – 4xy + 4y2
Answer: (x – 2y)2 = x2 – 4xy + 4y2.
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Example 2: Factorize x2 – 6xy + 9y2 by using a minus b whole square formula.
Solution:
To factorize: x2 – 6xy + 9y2. We can rearrange the given expression as: x2 – 6xy + 9y2 = (x)2 – 2 (x) (3y) + (3y)2. Using (a – b)2 formula: a2 – 2ab + b2 = (a – b)2 Substitute a = x and b = 3y in this formula: (x)2 – 2 (x) (3y) + (3y)2 = (x – 3y)2
Answer: x2 – 6xy + 9y2 = (x – 3y)2.
Example 3: Simplify the following using the (a – b)2 formula: (7x – 4y)2.
Solution:
a = 7x and b = 4y Using formula (a – b)2 = a2 – 2ab + b2 (7x – 4y)2 = (7x)2 – 2(7x)(4y) + (4y)2 = 49×2 – 56xy + 16y2.
Answer: (7x – 4y)2 = 49×2 – 56xy + 16y2.
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