Proving the Equation ( ab^2 2a^2b^2 ) for Real Numbers ( a ) and ( b )

When dealing with mathematical equations, it is crucial to understand the conditions under which certain equations hold true. Specifically, we aim to prove the equation ab2 2a2b2 for real numbers a and b. This involves a detailed exploration of the algebraic steps and conditions that lead to the proof.

Proof for ( a eq b )

First, let's consider the case where a ≠ b. We start with the expression:

0 (a - b)2

We know that the square of any real number is non-negative. Since (a - b)2 0, this implies that a - b 0. Therefore, a b.

Proof for ( a b )

Now, let's consider the case where a b. Substituting a for b in the equation ab2 2a2b2 yields:

a·a2 2a2·a2

Simplifying the right-hand side gives:

a3 2a?

This can be rewritten as:

a3 - 2a? 0

Factoring out a3 gives:

a3(1 - 2a) 0

Since a3 is zero when a 0, and the term 1 - 2a is zero when a 1/2, the equation holds for these values. However, the general equality holds when a b.

General Proof Using Algebraic Manipulation

To prove the equation comprehensively, we can rearrange and manipulate the initial expressions. Starting with the expression:

(a - b)2 0

We can expand and simplify both sides:

0 (2a2b2 - ab2)

This can be rewritten as:

2a2b2 - ab2 0

Factoring out ab2 on the left-hand side gives:

ab2(2a - 1) 0

This implies that either ab2 0 or 2a - 1 0. For the case where ab ≠ 0, we must have:

2a2b2 ab2

This leads to:

2a2b2 ab2

Thus, we conclude that the equation ab2 2a2b2 is valid for all real numbers a and b when a b.

The proof is concluded as follows:

Q.E.D.

Conclusion

We have proven the equation ab2 2a2b2 for real numbers a and b, with a detailed exploration of the algebraic steps and conditions. The proof demonstrates the importance of understanding the properties of real numbers and their manipulation in mathematical proofs.

For further reading and deeper exploration, one can refer to advanced algebra textbooks or online resources on mathematical proofs. Understanding these concepts can enhance one's problem-solving skills in mathematics and related fields.