Question

Find the condition on a , b, c are and d such that equations 2ax³ + bx² + cx + d = 0 and 2ax² + 3bx + 4c = 0 have a common root .

Answer 1

Hey !!!

Solution :-let α ' be a common root of the given equations , then 2aα^3 + bα^2 + cα + d =0

And , 2aα^3 + bα^2 + cα + d = 0

and 2α^2 + 3bα + 4c = 0

multiply (2) by (α) and substract (1) from it , to get

(2bα^2 + 3cα - d ) =0

Now , (2) and (3) are quadratic having a common root α , so

α^2/3bd -12c^2 = α/8bc + 2ad = 1/6ac -6b^2,

α^2 =bd +4c^2/2b^2-2ac , a =4bc + ab / 3ac -3b^2

Eliminating α from these two equations, we get

(4bc + ad)^2 = 9/2 (bd + 4c^2) (b^2 -ac),

which is required condition.

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Hope it helps you !!

@mr.Rajukumar111

Answer 2

Heya dude!!

Here's your answer

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let α be a common root of the given equations , then 2aα^3 + bα^2 + cα + d =0

And , 2aα^3 + bα^2 + cα + d = 0

and 2α^2 + 3bα + 4c = 0

multiply (2) by (α) and substract (1) from it , to get

(2bα^2 + 3cα - d ) =0

Now , (2) and (3) are quadratic having a common root α , so

α^2/3bd -12c^2 = α/8bc + 2ad = 1/6ac -6b^2,

α^2 =bd +4c^2/2b^2-2ac , a =4bc + ab / 3ac -3b^2

Eliminating α from these two equations, we get

(4bc + ad)^2 = 9/2 (bd + 4c^2) (b^2 -ac),

which is required condition.

_______________________________

Hope it helps you dear!! :)

# Devil king