- #1
Mr Davis 97
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I am given the following problem: Find all ordered pairs (a, b) such that ##2a + b = 12## and ##ab = 3##. Given this system of equations, I simply use substitution, and then solve the quadratic ##2a^2 - 12a + 3 = 0##. Solving this, I obtain two ordered pairs: ##\displaystyle (\frac{6 + \sqrt{30}}{2},~6 - \sqrt{30})## and ##\displaystyle (\frac{6 - \sqrt{30}}{2},~6 + \sqrt{30})##. These are all of the ordered pairs that I found. The problem asks to find all of the ordered pairs that satisfy the system. How can I be absolutely sure (through some kind of informal logic) that only two such ordered pairs exist, and these two are those ordered pairs?