Question

if one root of the equation ax²+bx+c=0 is the cube of the other, prove that,(b²-2ac)²=ca(c+a)²​

Answer 1

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Quadratic Equations

Nature of Roots

If one root of the equation...

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Asked on December 30, 2019 by

Zilvi Sindhu

If one root of the equation ax

2

+bx+c=0;a,b,c∈R is cube of other, then

a+c

b

2

−2ac

is equal to

HARD

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ANSWER

Let the roots be α and α

3

∴α+α

3

=−

a

b

and α

4

=

a

c

⇒α

2

(1+α

2

)

2

=

a

2

b

2

⇒α

2

+2α

4

+α

6

=

a

2

b

2

⇒

a

c

+

a

2c

+

a

c

a

c

=

a

2

b

2

⇒

a

c

(1+

a

c

)=

a

2

b

2

−2ac

⇒

a

c

(

a

a+c

)

2

=

a

4

(b

2

−2ac)

2

⇒ac=(

a+c

b

2

−2ac

)

2

∴

a+c

b

2

−2ac

=±

ac