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State whether the following statement is true or false.

The quadratic polynomial $x 2 + k x + k$ has equal zeroes for some odd integer $k > 1$ .

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True

False

answer is B.

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Detailed Solution

Given quadratic polynomial is,

x 2 + k x + k

We know that the roots of the quadratic equation

a x 2 + b x + c = 0

are equal if

b 2 − 4 a c = 0

.
On comparing given polynomial with

a x 2 + b x + c = 0

we get,

a = 1, b = k

and

c = k

Then,

⇒ k 2 − 41 k = 0 ⇒ k k − 4 = 0 ⇒ k = 0 o r (k − 4) = 0 ⇒ k = 0 o r k = 4

The possible value of k is 0 or 4 for the roots of the quadratic equation

x 2 + k x + k = 0

to be equal.
The quadratic polynomial

x 2 + k x + k

cannot have equal zeroes for some odd integer

k > 1

.
Therefore, the given statement is false.
Hence, option (2) is correct.

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