Total number of integral values of $a$ such that $x^{2} + a x + a + 1 = 0$ has integral roots is

1

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2

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3

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4

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Solution

The correct option is B $2$
$f o r a e q u a t i o n {a x}^{2} + b x + c = 0 t o h a v e i n t e g e r r o o t s c o n d i t i o n s a r e a = 1, b \in I, c \in I a n d b^{2} - 4 a c i s a p e r f e c t s q u a r e x^{2} + a x + a + 1 = 0 a \in I ∴ a + 1 \in I a^{2} - 4 (a + 1) i s p e r f e c t s q u a r e ∴ a^{2} - 4 (a + 1) i s p e r f e c t s q u a r e ∴ a^{2} - 4 a - 4 = p^{2} a^{2} - 4 a - 4 - p^{2} = 0 {(a - 2)}^{2} - 8 - p^{2} = 0 {(a - 2)}^{2} = p^{2} + 8 ∴ p^{2} = 1 {(a - 2)}^{2} = 9 {(a - 2)}^{2} = 3^{2} ∴ a - 2 = 3 o r a - 2 = - 3 a = 5 o r a = - 1 ∴ 2 v a l u e s o f a$

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