If x2+5x+6=0 and 2x2+ax+b=0 has both roots in common then the value of a+b is __
Answers
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
Given : x²+5x+6=0 and 2x²+ax+b=0 has both roots in common
To find : Value of a + b
Solution:
x²+5x+6=0 and 2x²+ax+b=0
has both roots in common
=> both Equations are same
=> 1/2 = 5/a = 6/b
=> a = 10 & b = 12
=> a + b = 10 + 12 = 22
other method
both roots are same hence sum of Zeroes & product of Zeroes would also be common
=> -5 = - a/2 & 6 = b/2
=> a = 10 & b = 12
=> a + b = 10 + 12 = 22
value of a+b is 22
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