Question

if a + b +c = 5 and ab+bc+ca= 10, then prove that a^3+b^3+c^3− abc3= −25

Answer 1

Given:

a + b + c = 5

ab + bc + ac = 10

To Prove:

a³ + b³ + c³ - 3abc = -25

Method of Solution:

1. We can solve it by just using a formula,

a³ + b³ + c³ - 3ab

= (a+b+c)(a²+ b² + c²- ab - bc - ac)

2. But 1st we should find the value of a²+ b² + c².

So, Square both side get the required value.

3. Substitute the values and prove.

Solution:

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

or, 5² = a² + b² + c² + 2(10)

or, 25 = a² + b² + c² + 20

or, 25 - 20 = a² + b² + c²

or, 5 = a² + b² + c²

Now, We know the value of a² + b² + c²,

Again,

a³ + b³ + c³ - 3ab

= (a+b+c)(a²+ b² + c²- ab - bc - ac)

=(5){5 -(ab +bc +ac)}

= 5(5-10)

= 5(-5)

= -25

Hence, a³ + b³ + c³ - 3abc = -25 [Proved]