Question

if alpha and beta are the roots of the equation 4x²-5x-1=0, find the quadratic equation whose roots are: (a)2- a and 2-b, (b)a² and b², (c) ¹/a and ¹/b, (d) ²/a² and ²/b². note : a=alpha, b=beta​

Answer 1

Answer:

4x

2

−5x+2=0

Here, a=4,b=−5,c=2

α+β=

a

−b

=−

4

−5

=

4

5

and αβ=

a

c

=

4

2

=

2

1

Roots are α+

a

1

and β+

β

1

α+

α

1

+β+

β

1

=α+β+

α

1

+

β

1

=α+β+

αβ

β+α

=

4

5

+[

4

5

÷

2

1

]

=

4

5

+

2

5

=

4

15

....(1)

(α+

α

1

)(β+

α

1

)=αβ+

β

α

+

α

β

+

αβ

1

=αβ+

αβ

1

+

αβ

α

2

+β

2

α

2

+β

2

=(α+β)

2

−2αβ

=(

4

5

)

2

−2(

2

1

)=

16

25

−1=

16

9

∴αβ+

αβ

1

+

αβ

α

2

+β

2

=

2

1

+[1÷

2

1

]+

16

9

÷

2

1

∴x

2

−

4

15

x+

8

29

=0 ... from (1) and (2)

∴8x

2

−30x+29=0 (multiplying by 8)

Hence the required equation is 8x

2

−30x+29=0