Permutations And Combinations


If A and G are respectively the A.M. and G.M. between two positive numbers a and b, then proof the quardratic equation having a, b as it roots is straight x squared minus 2 Ax plus straight G squared equals 0.


Since A is the A.M. between a and b

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#6 {main}</pre>                                                                ...(i)
Since G is the G.M. between a and b    

∴                                  G = space space square root of ab                                                  ...(ii)                                               
From (i),         a + b = 2A

∴              Sum of the roots = 2A = S (Say)
From (ii), ab = G2

∴        Product of the roots = G2 = P (say)
Required quadratic equation having a and b as its roots is straight x squared minus Sx plus straight p equals 0 
or     straight x squared minus 2 Ax plus straight G squared space equals 0

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