An algebra problem by Bala vidyadharan
Algebra
Level
pending
If and the equation (x-a)(x-10)+1=0 has integral roots ,then value of 'a' are
12,8
12,6
12,10
10,8
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1 solution
We have the quadratic equation x 2 − ( a + 1 0 ) x + ( 1 0 a + 1 ) = 0 , which has roots:
x = 2 ( a + 1 0 ) ± ( a + 1 0 ) 2 − 4 ( 1 ) ( 1 0 a + 1 ) = 2 ( ( a + 1 0 ) ± ( a − 1 0 ) 2 − 4 .
If x has intergral roots, then the discriminant must be a perfect square:
( a − 1 0 ) 2 − 4 = N 2 ⇒ [ ( a − 1 0 ) + N ] [ ( a − 1 0 ) − N ] = 2 2 ;
and if a ∈ Z then we require:
( a − 1 0 ) + N = 2 , ( a − 1 0 ) − N = 2 ⇒ a = 1 2 AND ( a − 1 0 ) + N = − 2 , ( a − 1 0 ) − N = − 2 ⇒ a = 8 .