1.

If x2+5x+6=0 and 2x2+ax+b=0 has both roots in common then the value of a+b is __

Answer»

Answer:

22

Step-by-step explanation:

⇒ x^2 + 5x + 6 =0

⇒ x^2 + ( 3 + 2 )x + 6 = 0

⇒ x^2 + 3x + 2x + 6 = 0

⇒ x( x + 3 ) + 2( x + 3 ) = 0

⇒ ( x + 3 ) ( x + 2 ) = 0

⇒ x = - 3 or - 2

  Hence roots of x^2 + 5x + 6 = 0 are - 3 and - 2. It means roots of 2x^2 + AX + b = 0 also are also - 3 and - 2.

Therefore, equation must satisfy the condition when x = - 3 and - 2.

When, x = - 3  

⇒ 2x^2 + ax + b = 0

⇒ 2( - 3 )^2 + a( - 3 ) + b = 0

⇒ 2( 9 ) - 3a + b = 0

⇒ 18 - 3a + b = 0

⇒ b - 3a = - 18                  ...( 1 )

 When x = - 2

⇒ 2( - 2 )^2 + a( - 2 ) + b = 0

⇒ 2( 4 ) + - 2a + b = 0

⇒ 8 - 2a + b = 0

⇒ b - 2a = - 8              ...( 2 )

 

    Subtracting ( 1 ) from ( 2 )

⇒ b - 2a - ( b - 3a ) = - 8 - ( - 18 )

⇒ b - 2a - b + 3a = - 8 + 18

⇒ a = 10

        Hence,

        b - 2a = - 8

        b = - 8 + 2( 10 )

        b = 20 - 8

        b = 12

Therefore,

⇒ a + b = 10 + 12

             = 22


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