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A mathematical proof serves as a comprehensive and logical argument that confirms the truth of a statement. In this exercise, we are required to prove the claim that \((a+b)^{2}=a^{2}+b^{2}\) if and only if \(a=0\) or \(b=0\). This phrase "if and only if" signifies a bi-directional proof, meaning we must prove both directions of the implication.

• "If direction" means we assume \(a=0\) or \(b=0\) and show \((a+b)^2 = a^2 + b^2\).
• "Only if direction" implies assuming \((a+b)^2 = a^2 + b^2\) and concluding that \(a=0\) or \(b=0\).

Breaking the problem into these two parts ensures that the condition is both necessary and sufficient. This methodical approach helps solidify the claim by addressing all possibilities and ensuring that no alternative is left unexplored.
Starting your proof, consider what makes one direction hold true, then tackle the reverse, showing without contradiction or possibility of error that each follows logically. This trains us not only in proving mathematical statements but in precise logical thinking.

Quadratic expansion refers to the process of multiplying binomial expressions to shape them into a quadratic form. It's pivotal in many algebraic problems. For instance, the expansion of \((a + b)^2\) is a classic example:\[(a + b)^2 = a^2 + 2ab + b^2\]This is derived by applying the distributive property twice. Consider:

• First, distribute \(a + b\) into itself: \((a + b)(a + b)\).
• Then multiply each term: \(a(a + b) + b(a + b)\).
• Finally, expand each part: \(a^2 + ab + ba + b^2\), which simplifies to \(a^2 + 2ab + b^2\).

Here, the middle term \(2ab\) is a result of the addition of two products: \(ab + ba\). It's crucial because it often brings out the hidden relationships between the components of the binomial. By understanding how expansion unfolds, solving various algebraic equations becomes much more manageable as you can effortlessly rearrange and simplify expressions.

Algebraic manipulation involves using algebraic properties to rearrange and simplify expressions. In this exercise, it plays a significant role by allowing us to equate and translate expressions that appear similar but are differently organized.
Once we establish that \((a + b)^2 = a^2 + b^2\), we can conduct algebraic manipulation to investigate the terms:

• We start with the expanded form \(a^2 + 2ab + b^2\).
• We equate it to \(a^2 + b^2\), seeing them as equal.
• Next, subtraction is used: \(a^2 + 2ab + b^2 - a^2 - b^2 = 2ab = 0\).

This subtraction simplifies to \(2ab = 0\). Recognizing that 2 can't be zero, we deduce \(a = 0\) or \(b = 0\) because the multiplication rule of zero states that having any nonzero factor multiplying to zero mandates at least one of the other factors must be zero.
This subtle rearranging and canceling approach vividly shows how each step logically follows from the last, creating a clear pathway from the hypothesis to the conclusion.