SOLUTION: If a^2+b^2=65 and 4ab=24, calculate the value of A. (a+b)^2 B. (2a-2b)^2 Also, can you please show the solution to get the value of a and b?Thank you vey much!

Algebra ->  Equations -> SOLUTION: If a^2+b^2=65 and 4ab=24, calculate the value of A. (a+b)^2 B. (2a-2b)^2 Also, can you please show the solution to get the value of a and b?Thank you vey much!      Log On


   

Question 1086791: If a^2+b^2=65 and 4ab=24, calculate the value of
A. (a+b)^2
B. (2a-2b)^2
Also, can you please show the solution to get the value of a and b?Thank you vey much!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
%28a%2Bb%29%5E2=a%5E2%2B2ab%2Bb%5E2
So,
%28a%2Bb%29%5E2=%28a%5E2%2Bb%5E2%29%2B2ab
%28a%2Bb%29%5E2=%2865%29%2B%2824%2F2%29
%28a%2Bb%29%5E2=65%2B12
%28a%2Bb%29%5E2=77
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%282a-2b%29%5E2=4a%5E2-8ab%2Bb%5E2
%282a-2b%29%5E2=4%28a%5E2%2Bb%5E2%29-2%284ab%29
Plug in the values and calculate.
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4ab=24
ab=6
a=6%2Fb
Substitute,
%286%2Fb%29%5E2%2Bb%5E2=65
36%2Fb%5E2%2Bb%5E2=65
36%2Bb%5E4=65b%5E2
b%5E4-65b%5E2%2B36=0
Use a substitution,
x=b%5E2
x%5E2=b%5E4
x%5E2-65x%2B36=0
Complete the square,
x%5E2-65x%2B36=0
%28x%5E2-65x%2B%2865%2F2%29%5E2%29=%2865%2F2%29%5E2-36
%28x-65%2F2%29%5E2=4225%2F4-144%2F4
%28x-65%2F2%29%5E2=4081%2F4
x-65%2F2=0+%2B-+sqrt%284081%29%2F2
x=65%2F2+%2B-+sqrt%284081%29%2F2
x=%2865%2B-sqrt%284081%29%29%2F2
So then when x=%2865%2Bsqrt%284081%29%29%2F2
b%5E2=%2865%2Bsqrt%284081%29%29%2F2
b=sqrt%28%2865%2Bsqrt%284081%29%29%2F2%29 and b=-sqrt%28%2865%2Bsqrt%284081%29%29%2F2%29
You can then find a since,
a%5E2%2Bb%5E2=65
a%5E2=65-b%5E2
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Then do the same when x=%2865-sqrt%284081%29%29%2F2
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