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Which of the subsequent is correct for the zeroes of the quadratic polynomial $p (x) = x 2 + k x + k, k ≠ 0$ ?

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They can’t be positive

They can’t be negative

They are always equal

They are always unequal

answer is A.

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Detailed Solution

The given polynomial is,

p (x) = x 2 + k x + k, k ≠ 0

⇒ α + β = − k & α β = k

α and β

are the zeroes of

a x 2 + b x + c,

then

α + β = − b a and α β = c a

Since the given polynomial is,

p (x) = x 2 + k x + k, k ≠ 0

⇒ α + β = − k & α β = k

The sum and product of two numbers will be positive if both numbers are positive.
Therefore

α and β

can’t be positive, as if both are positive then sign of

α + β

and

α β

will be positive but here clearly they are of opposite sign.
Therefore, they can’t be positive.
If

k > 0,

then both roots can be negative.
If the discriminant is zero, then we have,

k 2 − 4 k = 0 ⇒ k = 4

Since it is given that, k

≠ 0.

So both roots can be equal.
Also, it will have equal zeroes only if

k = 4.

Therefore, they aren’t always unequal.
Thus, the zeroes of the quadratic polynomial

p (x) = x 2 + k x + k, k ≠ 0

can’t be positive.
Hence, option (1) is correct.

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