Special Product A^2 - B^2 and A^2 + B^2

The product format A² - B² is a special form of algebraic expression.

It is equal to (A + B)(A - B).

Many other expressions, like the cos2A = cos²A - sin²B,
can also be expressed in the special product form, that is,

cos2A = (cos A + sin A)(cos A - sin A).

But, how about A² + B² ?

Can it be expressed in the (A + B)(A - B) format?

Why not.

However we need to deviate a bit from the norm.

We need to know the imaginary "i" in complex number system.
Click here for a review to it.

Since i² = -1, we can make use of this property for the special product A² + B².

Here it goes....

A² + B² = A² - (i²)(B²)

Using the basic of Indices, it can be modified to,

A²-(iB)²

This becomes, therefore, (A + iB)(A - iB).

In summary,

A² - B² = (A + B)(A - B), and

A² + B² = (A + iB)(A - iB).

The principles of the special product still holds regardless of the addition or subtraction operation between A square and B square.

Student: What an interesting twist is mathematics in this matter. Anything seems to be simple if we know the technique!

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