Proving a b Given a Specific Algebraic Equation

Today, we'll explore how to prove the equality a b given the algebraic equation:

a b2 2a2 – 2b2

Step 1: Expand and Simplify the Left-Hand Side

First, we'll expand the left-hand side of the given equation:

a b2 a2 – 2ab b2

Step 2: Write the Equation and Rearrange Terms

Next, we can rewrite the original equation with the expanded left-hand side and place all terms on one side of the equation:

a2 – 2ab b2 2a2 – 2b2

After rearranging, we get:

a2 – 2ab – b2 – 2a2 – 2b2 0

The equation simplifies to:

-a2 – 2ab – b2 0

Step 3: Factor the Equation

Now we factor the left-hand side of the equation:

-a2 – 2ab – b2 - (a2 2ab b2)

This can be further factored as:

- (a b2)2 0

Step 4: Apply the Property of Squares

Since a square of any real number is zero only when the number itself is zero:

(a b) 0

Therefore, we conclude:

a b

Alternative Geometric Proof

Another way to approach this problem is through a geometric proof. Let's consider a square with side length b:

The area of the square can be expressed as:

Area of large square ab2

Let's dissect this square into smaller shapes:

Four smaller squares (each with side a) and Two rectangles (each with dimensions a by b).

The total area can be computed as:

Area of large square a2 2ab b2

Setting the two expressions for the area of the large square equal to each other:

ab2 a2 2ab b2

Rearranging this, we get:

ab2 a2 2ab b2

simplifies to:

ab2 a2 – 2ab b2

Which further simplifies to:

a b

Special Considerations

It is important to note that this equality holds true only for numbers like integers, real numbers, or variables representing such numbers. In the context of non-commutative rings, particularly matrix algebra, the equality might not hold as the operations do not necessarily commute.

For example, if:

ab2 abab aba – abb

This simplifies to:

a2 – ba – ab – b2

For this to be equal to 2a2 – 2b2, the following must hold:

ba ab

Which is true only for certain matrices, such as diagonal matrices or matrices in a commutative field.

Conclusion

We have demonstrated two methods for proving the equality a b given the equation a b2 2a2 – 2b2. The algebraic approach involves simplifying and factoring the equation, while the geometric approach uses the concept of square and rectangle areas.