Binomial formulas - Calculate equations

Mathematicians introduced the well-known three binomial formulas to make it easier to calculate certain equations. Three formulas that cause a lot of headaches instead of making things easier, especially for students. But as the saying goes? All beginnings are difficult and, as we all know, practice makes perfect.

Someone who is familiar with bracket calculations does not really need the three binomial formulas at all. If you are familiar with the laws of arithmetic, it becomes clear that all three formulas inevitably follow from these laws. But why are the binomial formulae even taught at school? Quite simply, they make life easier for every mathematician in the form of a shortcut.

In order to be able to understand the binomial formulas, a knowledge of bracket calculation is necessary. This includes

• Multiplying out brackets
• Dot before bracket or dash

The first binomial formula

In the first binomial formula, multiplying out the parenthesis is important. As a result, the first binomial formula is as follows:

( x y ) ² = x² 2xy y²

The derivation: ( x y ) ² = ( x y ) - ( x y ) = x² xy yx y² = x² 2xy y²

The derivation is particularly helpful when it comes to the question of where the formula comes from. It shows in the simplest way how the multiplication of the parenthesis works. To illustrate this, here are two examples to help you understand the first binomial formula.

1. Example: ( 5 6 ) ² = 52 2 - 5 - 6 62 = 25 60 36 = 121
2. Example: ( 8 9 ) ² = 82 2 - 8 - 9 92 = 64 144 81 = 289

A little tip on the side: When looking at the binomial formula, think about what x and y are. The numbers for x and y can then be used. To clarify this, compare the first binomial formula mentioned above with the two examples below.

The second binomial formula

The second binomial formula has a similar structure to the first, except that the plus is replaced by a minus. The second binomial formula is therefore composed as follows:

( x - y )² = x² - 2xy y²

The derivation: ( x - y ) ² = ( x - y ) - ( x - y ) = x² - xy - yx y² = x² - 2xy y²

Ultimately, this is about the difference in the existing parenthesis. This should be recognized on the one hand and inserted on the other. Here are two explicit examples to illustrate this.

1. Example: ( 7 - 3 ) ² = 72 - 2 - 7 - 3 32 = 49 - 42 9 = 16
2. Example: ( 8 - x ) ² = 82 - 2 - 8 - x x² = 64 - 16x x²

The third binomial formula

In the third and last binomial formula, a total of two brackets are multiplied together, resulting in the following formula:

( x y ) - ( x - y ) = x² - y²

The derivation: ( x y ) - ( x - y ) = x² - xy yx - y² = x² - y²

The third binomial formula is only used in the case of two brackets. It is important to note that only the variable in the second bracket changes. To illustrate the procedure for a third binomial formula, here are two examples.

1. Example: ( x 4 ) - ( x - 4 ) = x² - 42 = x² - 16
2. Example: ( 7 y ) - ( 7 - y ) = 72 - y² = 49 - y²