MHB Difference in expansion of brackets

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The expression \(x^2 + y^2 = 14\) cannot be accurately rewritten as \((x+y)^2 = 14\) because this transformation leads to the incorrect equation \(x^2 + 2xy + y^2 = 14\). The distinction between \((a+b)^2\) and \(a^2 + b^2\) is critical, as the former includes the cross term \(2ab\). This common mistake among students is referred to as "The Freshman's Dream." Understanding this difference is essential for correctly expanding and simplifying algebraic expressions. Misapplying these principles can lead to significant errors in problem-solving.
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Let's say that there is an expression $x^2+y^2=14$

Cannot this expression be written as $(x+y)^2 = 14$ taking out the square which is common to both the terms $x$ and $y$ but after writing like that doesn't this become $x^2+2xy+y^2$=14

So writing the expression like that would it remain valid or not ?
 
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In general, we have:

$$(a+b)^2\ne a^2+b^2$$

Students make this mistake so often it has been given a name...The Freshman's Dream.
 
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