Question

If α,β are roots of eqn x^2−2x+3=0 then find the an eqn whose roots are α^3−3α^3+5α−2,β^3−β^2+β+5​

Answer 1

Explanation:

Answer

Given α & β are the roots of equation x

2

−2x+3=0

To find: Equation whose roots are α

3

−3α

2

+5α−2,

β

3

−β

2

+β+5

Sol: x

2

−2x+3=0

x=

2

2±

4−12

x=

2

2±

8i

x=1±

2

i

α=1+

2

i, β=1−

2

i

α

3

−3α

2

+5α−2=(1+

2

i)

3

−3(1+

2

i)

2

+5(1+

2

i)−2

=1+2

2

i

3

+3

2

i(1+

2

i)−3(1+2i

2

+2

2

i)+5+5

2

i−2

=1−2

2

i+3

2

i−6−3+6−6

2

i+3+5

2

i

=1+

2

i−

2

i

=1

β

3

−β

2

+β+5=(1−

2

i)

3

−(1−

2

i)

2

+(1−

2

i)+5

=1−(

2

i)

3

−3

2

i(1−

2

i)−(1+2i

2

−2

2

i)+(1−

2

i)+5

=2

∴ Equation whose roots are α

3

−3α

2

+5α−2 and β

3

−β

2

+β+5 is (x−1)(x−2)=0

⇒x

2

−3x+2=0

∴ x 2

−3x+2=0.