Question

If a + b + c = 5 and ab + bc + ca = 10, then prove that a3 + b3 + c3 - 3abc = -25.

Answer 1

Given:

a+b+c = 5

ab +bc +ca = 10

To prove:

a³ + b³ + c³ -3abc = -25

Solution:

We know that,

(a+b+c)² = a² +b² +c² +2(ab+bc+ca)

Now we will put the given values

=> (5)² = a² +b² +c² +2(10)

=> 25 = a² +b² +c² +20

=> a² +b² +c² = 25-20

=> a² +b² +c² = 5

We also know that

a³+b³+c³-3abc = (a+b+c)(a²+b²+c²-ab-bc-ca)

=> a³+b³+c³-3abc = (a+b+c)[a²+b²+c²-(ab+bc+ca)]

Again substituting values

=> a³+b³+c³-3abc = (5) [ 5-(10) ]

=> a³+b³+c³-3abc = 5× (5-10)

=> a³+b³+c³-3abc = 5×(-5)

=> a³+b³+c³-3abc = -25

Hence proved

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