A B whole square may be taken as A minus B whole square or A plus B whole square. The formulas for a – b whole square and a + b whole square are used to expand the binomials in algebra. We can say that these are referred to as the square of a binomial expansion. (a – b)2 and (a + b)2 are the basic algebraic identities used in the simplification of numerical expressions as well as binomial expressions with variables. In this article, you will understand the a minus b whole square formula and a plus whole formula along with proof and examples in detail.
Learn: Algebraic identities
A Minus B Whole Square
Let’s learn what is a minus b whole square formula here.
(a – b)2 is equal to a2 – 2ab + b2
(a – b)2 formula is used to expand the binomial term when squared, and to find the square of the difference between two numbers.
The formula for (a – b)2 can be derived as:
(a – b)2 = (a – b)(a – b)
= (a)(a) – (a)(b) – (b)(a) + (b)(b)
= a2 – 2ab + b2
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(a – b)2 = a2 – 2ab + b2 |
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A – B Whole Square Formula Geometric Proof
Let us consider a square whose side length is “a” such that its area is equal to “a2”.
Now, take a small square from this whose side is (a – b) as shown in the below figure.
Thus, we get two rectangles – one is horizontal and the other is vertical.
Let’s add the areas of all these three quadrilaterals.
Area of square with side (a – b) = (a – b)2
Area of vertical strip (rectangle) = Length × Breadth = a × b = ab
Area of horizontal strip (rectangle) = Length × Breadth = (a – b) × b = b(a – b)
Area of square with side “a” = Area of square with side (a – b) + Areas of two rectangles
a2 = (a – b)2 + ab + b(a – b)
a2 = (a – b)2 + ab + ab – b2
a2 = (a – b)2 + 2ab – b2
Therefore, (a – b)2 = a2 – 2ab + b2
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Watch the Video below, to Know about A minus B Whole Square Formula
A Plus B Whole Square
(a + b)2 formula is one the most commonly used algebraic identity in maths. Let’s learn what is a plus b whole square formula here.
(a + b)2 is equal to a2 + 2ab + b2
The a + b whole square formula is used to expand the binomial term when squared, and to find the square of the difference between two numbers.
The formula for (a – b)2 can be derived as:
(a + b)2 = (a + b)(a + b)
= (a)(a) + (a)(b) + (b)(a) + (b)(b)
= a2 + 2ab + b2
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(a + b)2 = a2 + 2ab + b2 |
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A Plus B Whole Square Formula Proof
A plus B whole square formula can be proved as we did for a – b whole square. In the case of a – b whole square, we reduced the length of the original square, whereas for a plus b whole square we will be extending the square length by b units.
First, consider a square with side “a” such that its area will be “a2”.
Now, extend the lengths of this square by b units as shown in the below figure such that we get two rectangles and a small square with side b.
Side of the new (big) square = a + b
Area of bigger square = (a + b)2
Area of vertical strip (rectangle) = Length × Breadth = a × b = ab
Area of horizontal strip (rectangle) = Length × Breadth = a × b = ab
Area of smaller square = b2
Area of square with side (a +b) = Area of square with side “a” + Areas of two rectangles + Area of small square
(a + b)2 = a2 + ab + ab + b2
Therefore, (a + b)2 = a2 + 2ab + b2
Solved Examples
Example 1:
Evaluate 1082 using (a + b)2 formula.
Solution:
Let’s write 108 as: 108 = 100 + 8
1082 = (100 + 8)2
Using the formula (a + b)2 = a2 + 2ab + b2,
1082 = (100)2 + 2(100)(8) + (8)2
= 10000 + 1600 + 64
= 11664
Example 2:
Expand (3x – 7y)2 using a minus b whole square formula.
Solution:
(3x – 7y)2
Let a = 3x and b = 7y
Using the formula (a – b)2 = a2 – 2ab + b2,
(3x – 7y)2 = (3x)2 – 2(3x)(7y) + (7y)2
= 9x2 – 42xy + 49y2
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