One Two Four

Level pending

$a+b+c=0$ and

$a^2+b^2+c^2=10$ ,

then what is the value of $a^4+b^4+c^4$ ?

1 solution

Umes Stha
Aug 5, 2019

we have,

a + b + c = 0 ------------equation (i)

and

a² + b² + c² = 10 --------equation (ii)

we know,

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

Substituting respective values from equations (i) & (ii) to above equation, we get

0² = 10 + 2 (ab + bc+ca)

ab + bc+ca = -5

On squaring, we get

(ab + bc+ca)² = (-5)²

(ab)² + (bc)² + (ca)² + 2 (ab.bc + bc.ca + ca.ab) = 25 (Applying equation (iii) i.e. (a + b + c)² = a² + b² + c² + 2 (ab + bc+ca))

a²b² + b²c² + c²a² + 2 { abc (a + b + c) } = 25

a²b² + b²c² + c²a² + 2 {abc * 0 } = 25 (From equation (i) )

a²b² + b²c² + c²a² = 25

Taking equation (ii) and squaring it, we get

(a² + b² + c² )² = 10²

(a²)² + (b²)² + (c²)² + 2 (a².b² + b².c² + c².a²) = 100 (Again applying equation (iii) i.e. (a + b + c)² = a² + b² + c² + 2 (ab + bc+ca))

a⁴ + b⁴ + c⁴ + 2 (25) = 100 (From above substituting value of a²b² + b²c² + c²a² )

Hence,

a⁴ + b⁴ + c⁴ = 50