SOLUTION: Plz solve this... 1) If the arithematic mean between two numbers is twice there geometric mean then prove that the ratio of 2 nos. is (2+sqrt3):(2-sqrt3)

Question 73945: Plz solve this...
1) If the arithematic mean between two numbers is twice there geometric mean then prove that the ratio of 2 nos. is (2+sqrt3):(2-sqrt3)
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
If the arithmetic mean between two numbers is twice their geometric mean then prove that the ratio of 2 nos. is (2+sqrt3):(2-sqrt3)
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Let a,b be the two numbers:
arithmetic mean = (a+b)/2
geometric mean = sqrt(ab)
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EQUATIONS:
(a+b)/2 = 2(sqrt(ab))
a+b = 4sqrt(ab)
square both sides to get:
(a+b)^2 = 16ab
a^2-14ab+b^2=0
Use the Quadratic formula to solve for "a" in terms of "b":
a=[14b+-sqrt((14b)^2-4b^2)]/2
a=[14b+-sqrt(192b^2)]/2
a=[14b+-4b(sqrt3)]/2
a = [7 +- (2sqrt3)b]
a:b = (7+2sqrt(3)):1
or a:b =(7-2sqrt(3)):1

Cheers,
Stan H.

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