Some Number Problems

IF it is given that a+b=c+d AND ab=cd, does it imply that a=c and b=d or a=d and b=c?

#ProofTechniques

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

Squaring the 1st expression and subtracting the 2nd from the 1st and rearranging leads to the result: (a+c)(a-c) = (d+b)(d-b) But we already know that a-c = d-b, so we have (a+c) = (d+b), or a = d. Similarly for a = c, etc.

Michael Mendrin - 7 years, 4 months ago

Thankyou very much Sir! I researched about it and found other ways to prove it as well.. One would be to assume they are the roots of a polynomial, they should be equal because they would be the roots of the same polynomial. Thanks for you time again! :)

Keshav Gupta - 7 years, 4 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$