Question

If m and n are the roots (not necessarily real) of the quadratic equation x2 + 2x + 3 =0 , find the value of m^3 + n^3.

Answer 1

m^3 + n^3 = 10..........

Answer 2

Given m and n are roots of quadratic equation

sum of roots = m + n

also

sum of roots = – b / a = – 2 / 1 = – 2

▶ m + n = – 2

Product of roots = m × n

also product of roots = c / a = 3 / 1 = 3

▶ mn = 3

( m + n )² = m² + n² + 2 mn

( – 2 )² = m² + n² + 2 × 3

4 – 6 = m² + n²

– 2 = m² + n² – ( 1 )

Now,

m³ + n³ = ( m + n ) ( m² – mn + n² )

m³ + n³ = ( – 2 ) ( m² + n² – 3 )

m³ + n³ = ( – 2 ) ( – 2 – 3 ) { from 1 }

m³ + n³ = ( – 2 ) ( – 5)

m³ + n³ = 10

Hope the solution is clear to!!