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Answer by Asinomás for There is only one distinct pair of numbers that...
If two numbers add to $a$ then you can write them as $\frac{a-x}{2}$ and $\frac{a+x}{2}$. The product is therefore $\frac{a^2-x^2}{4}$, clearly $x^2=y^2\iff x=\pm y$, but this only corresponds to...
View ArticleAnswer by mvw for There is only one distinct pair of numbers that multiply to...
You start with$$ x + y = c_1 \\x \, y = c_2$$and transform to$$y = c_1 - x \\x (c_1 - x) = c_2$$where the second equation can be transformed to$$0 = x^2 - c_1 x + c_2 = (x - c_1/2)^2 + c_2 - c_1^2/4...
View ArticleThere is only one distinct pair of numbers that multiply to a given number...
Statement:There is only one distinct pair of numbers that multiply to a given number and sum to a given number.Proof (almost complete):$$x+y=c_1 {}{}{}{} (1)$$$$xy=c_2 {}{}{}{} (2)$$We have from (1)...
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