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# Are the following statements 'True' or 'False'? Justify your answers.

The only value of $ k $ for which the quadratic polynomial $ k x^{2}+x+k $ has equal zeros is $ \frac{1}{2} $

Given:

The only value of \( k \) for which the quadratic polynomial \( k x^{2}+x+k \) has equal zeros is \( \frac{1}{2} \)

To do:

We have to find whether the given statement is true or false.

Solution:

Let $f(x) = kx^2 + x + k$

For equal roots, discriminant of $f(x)$ should be zero.

$D = b^2 - 4ac = 0$

Therefore,

$D=1^2-4(k)(k) = 0$

$1=4k^2$

$k^2=\frac{1}{4}$

$k =\sqrt{\frac{1}{4}}$

$k=\pm \frac{1}{2}$

So, for two values of $k$, the given quadratic polynomial has equal zeroes.

Hence, the given statement is false.

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