In this publication, we will consider one of the abbreviated multiplication formulas – the square of the difference. We will also analyze examples of solving problems to consolidate the material presented.
Difference squared formula
Difference squared a и b is equal to the square of the reduced (a) minus twice the product of the minuend and the subtrahend (a и b) plus the square of the subtrahend (b).
(a – b)2 = A2 – 2ab + b2
This expression is equivalent in reverse:
a2 – 2ab + b2 = (a – b)2
Proof of the formula
Since squaring is the multiplication of a number/expression by itself, let’s represent our formula as a multiplication of two identical brackets:
(a-b)(a-b).
It remains, according to the arithmetic rules, to remove the brackets, thereby decomposing the product into factors:
(a-b)(a-b) = a2 – ab – ba + b2 = A2 – 2ab + b2.
Examples of tasks
The task
Factor the square of the difference: (4x – 2y4)2.
Solution
Applying the abbreviated multiplication formula, we get:
(4x – 2y4)2 = (4x)2 – 2 ⋅ 4x ⋅ 2y4 + (2y4)2 = 16x2 – 16xy4 + 4y8
Note:
Knowing the formula also allows you to do quick mental calculations:
- 772 = (80 – 3)2 = 802 – 2 ⋅ 80 ⋅ 3 + 32 = 6400 – 480 + 9 = 5929.
- 482 = (50 – 2)2 = 502 – 2 ⋅ 50 ⋅ 2 + 22 = 2500 – 200 + 4 = 2304.