Question

verify the following: (a^2-b^2)(a^2+b^2)+(b^2-c^2)(b^2+c^2)+(c^2-a^2)(c^2+a^2)=0​

Answer 1

Solution :

LHS -

( a² - b²)( a² + b²) + (b² - c²)( b² + c²) + (c² + a²)( c² - a²)

=> a²( a² + b²) - b²( a² + b²) + b²( b² + c²) - c²( b² + c²) + c²( c² - a²) + a²( c² - a²)

=> a⁴ + a²b² - b²a² - b⁴ + b⁴ + b²c² - c²b² - c⁴ + c⁴ - c²a² + a²c² - a⁴

=> a⁴ + [ a²b² - b²a² ] - b⁴ + b⁴ + [ b²c² - c²b² ] - c⁴ + c⁴ - [c²a² - a²c² ]- a⁴

=> a⁴ - b⁴ + b⁴ - c⁴ + c⁴ - a⁴

=> 0

Hence Verified

__________________________

Additional Information :

(a + b)² = a² + 2ab + b²

(a + b)² = (a - b)² + 4ab

(a - b)² = a² - 2ab + b²

(a - b)² = (a + b)² - 4ab

a² + b² = (a + b)² - 2ab

a² + b² = (a - b)² + 2ab

2 (a² + b²) = (a + b)² + (a - b)²

4ab = (a + b)² - (a - b)²

ab = {(a + b)/2}² - {(a-b)/2}²

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

(a + b)³ = a³ + 3a²b + 3ab² b³

(a + b)³ = a³ + b³ + 3ab(a + b)

(a - b)³ = a³ - 3a²b + 3ab² - b³

a³ + b³ = (a + b)( a² - ab + b² )

a³ + b³ = (a + b)³ - 3ab( a + b)

a³ - b³ = (a - b)( a² + ab + b²)

a³ - b³ = (a - b)³ + 3ab ( a - b )

__________________________