Question

The two AM between -6and 14

Answer 1

The Arithmetic Mean (A.M) between two numbers exceeds their Geometric Mean (G.M.) by 22 and the GM exceeds the Harmonic Mean (H.M) by 1.61.6. Find the numbers.

My Attempt: Let the numbers be aa and bb. Then,
A.M=a+b2
A.M=a+b2
G.M=ab−−√
G.M=ab
H.M=2aba+b
H.M=2aba+b
According to question:
a+b2=ab−−√+2
a+b2=ab+2
a+b2−2=ab−−√
a+b2−2=ab
a+b−4=2ab−−√
a+b−4=2ab
Also,
ab−−√=2aba+b+1.6
ab=2aba+b+1.6
Then,
a+b−4=2(2aba+b+1.6)
a+b−4=2(2aba+b+1.6)
a+b−4=4ab+3.2(a+b)a+b
a+b−4=4ab+3.2(a+b)a+b
(a+b−4)(a+b)=4ab+3.2(a+b)
(a+b−4)(a+b)=4ab+3.2(a+b)
(a+b)2−4(a+b)=4ab+3.2a+3.2b
(a+b)2−4(a+b)=4ab+3.2a+3.2b
How do I solve further?