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  • Please,please help me - Complex numbers and quadratic equations - JEE Main

If  ax^{2}+bx+a= 0  and  x^{3}-2x^{2}+2x-1= 0  have two common roots then a+b  equals

  • Option 1)

    -3

  • Option 2)

    -2

  • Option 3)

    -1

  • Option 4)

    0

 

Answers (1)

ax^{2}+bx+a=0

Multiplying by x both sides

ax^{3}+bx^{2}+ax=0\cdots \cdots \left ( 1 \right )

x^{3}-2x^{2}+2x-1=0

Multiplying by a both sides

ax^{3}-2ax^{2}+2ax-a=0\cdots \cdots \left ( 2 \right )

(1) - (2) we get

\left (2a+b \right )x^{2}-ax+a=0 

This will also have both roots common with given two equations, so we can say

\left (2a+b \right )x^{2}-ax+a=0 & ax^{2}+bx+a=0 have both roots common, so

\frac{2a+b}{a}=\frac{-a}{b}=\frac{a}{a}\Rightarrow a+b=0

\therefore Option (D)

 

Nature of Common roots -

If  P_{1}\left ( x \right )= 0  and  P_{2}\left ( x \right )= 0 are two polynomial equations with same common root then any third polynomial equation obtained with algebra of first two will also have those common roots.

-

 

 


Option 1)

-3

This is incorrect

Option 2)

-2

This is incorrect

Option 3)

-1

This is incorrect

Option 4)

0

This is correct

Posted by

Vakul

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