Question

if a+b+c=6,a^2+b^2+c^2=14,a^3+b^3+c^3=36.Then find the value of abc

Answer 1

a + b +c =6                      --(eq.1)

a² + b² + c² = 14

a³ + b³ +c³ = 36

(a+b+c)² = a² +  b² + c² + 2ab + 2bc + 2ac

(6)² = (14) + 2(ab + bc + ca)

36 = 14 + 2(ab + bc + ca)

2(ab + bc + ca) = 36 -14 = 22

ab + bc + ca = 22 / 2 = 11

ab + bc + ca = 11             --(eq.2)

Multiplying (eq.1) and (eq.2),

(a + b + c)(ab + bc + ca) = 6 × 11

a²b + abc + ca² + ab² + b²c + abc + abc + bc² + c²a = 66

3abc + a²b + ca² + ab² + b²c + bc² + c²a  = 66

2abc + abc + a²b + ca² + ab² + b²c + bc² + c²a = 66

2abc + a²b + ca² + ab² + b²c + bc² + c²a = 66 - abc          --(eq3)

(a+b+c)³ = a³ + b³ + c³ + 3a²b + 3a²c + 3b²a + 3b²c + 3c²a + 3c²b + 6abc        --(eq.4)

Putting eq.3 in eq.4......
 
(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                      ---eq.5

Putting eq.3 in eq.5

60 = 66 - abc

abc = 66 - 60 = 6

abc = 6

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