a+b+c = 6, a^2 + b^2 + c^2 = 14 and a^3 + b^3 + c^3 = 36 then find abc
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a+b+c=6........................................1
a²+b²+c²=14.................................2
a³+b³+c³=36.................................3
We know that,
(a+b+c)²=a²+b²+c²+2ab+2bc+2ac
(a+b+c)²=a²+b²+c²+2(ab+bc+ac)
By putting eq1 and eq2 we get,
(6)²=14+2(ab+bc+ac)
(36-14)/2=ab+bc+ac
ab+bc+ac=11................................4
By multiplying eq1 and eq4 we get,
(a+b+c)(ab+bc+ac)=6×11
a²b+abc+a²c+b²a+b²c+abc+abc+c²b+c²a=66
a²b+a²c+b²a+b²c+c²a+c²b+3abc=66
a²b+a²c+b²a+b²c+c²a+c²b+2abc+abc=66
a²b+a²c+b²c+b²a+c²a+c²b+2ab=66-abc..................................................5
We also know that,
(a+b+c)³=a³+b³+c³+3a²b+3a²c+3b²a+3b²c+3c²a+3c²b+6abc
Putting eq1 and eq3 we get,
(6)³=36+3{a²b+a²c+b²a+b²c+c²a+c²b+2abc}
(216-36)=3{a²b+a²c+b²a+b²c+c²a+c²b+2abc}
60=a²b+a²c+b²a+b²c+c²a+c²b+2ab
putting eq5 in above equation we get,
60=66-abc
abc=66-60
abc=6
Hence value of abc=6.
a²+b²+c²=14.................................2
a³+b³+c³=36.................................3
We know that,
(a+b+c)²=a²+b²+c²+2ab+2bc+2ac
(a+b+c)²=a²+b²+c²+2(ab+bc+ac)
By putting eq1 and eq2 we get,
(6)²=14+2(ab+bc+ac)
(36-14)/2=ab+bc+ac
ab+bc+ac=11................................4
By multiplying eq1 and eq4 we get,
(a+b+c)(ab+bc+ac)=6×11
a²b+abc+a²c+b²a+b²c+abc+abc+c²b+c²a=66
a²b+a²c+b²a+b²c+c²a+c²b+3abc=66
a²b+a²c+b²a+b²c+c²a+c²b+2abc+abc=66
a²b+a²c+b²c+b²a+c²a+c²b+2ab=66-abc..................................................5
We also know that,
(a+b+c)³=a³+b³+c³+3a²b+3a²c+3b²a+3b²c+3c²a+3c²b+6abc
Putting eq1 and eq3 we get,
(6)³=36+3{a²b+a²c+b²a+b²c+c²a+c²b+2abc}
(216-36)=3{a²b+a²c+b²a+b²c+c²a+c²b+2abc}
60=a²b+a²c+b²a+b²c+c²a+c²b+2ab
putting eq5 in above equation we get,
60=66-abc
abc=66-60
abc=6
Hence value of abc=6.
prajapatyk:
if there is any doubt in this method i will clear
So u deserve to be the brainlist
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Answered by
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Answer:
and=6 please mark as brainliest
Step-by-step explanation:
x=3+2√2 then find√x-(1/√x)
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