Difference in expansion of brackets

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SUMMARY

The discussion centers on the mathematical expression $x^2 + y^2 = 14$ and the common misconception that it can be rewritten as $(x + y)^2 = 14$. This transformation is incorrect because $(x + y)^2$ expands to $x^2 + 2xy + y^2$, which introduces an additional term, $2xy$. This error is commonly referred to as "The Freshman's Dream," highlighting a frequent mistake among students in algebra.

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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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