Answer the following and justify:
Can the quadratic polynomial $ x^{2}+k x+k $ have equal zeroes for some odd integer $ k>1 $ ?

Given:

Quadratic polynomial \( x^{2}+k x+k \) and \( k>1 \).

To do:

We have to find whether the quadratic polynomial \( x^{2}+k x+k \) can have equal zeroes for some odd integer \( k>1 \).

Solution:

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

If $p(x)$ has equal zeroes, then its discriminant is zero.

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

$a =1, b = k$ and $c = k$

Therefore,

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

$k(k- 4)=0$

$k =0$ or $k=4$

This implies, the quadratic polynomial $p(x)$ has equal zeroes at $k =0, 4$.

Hence, the quadratic polynomial \( x^{2}+k x+k \) cannot have equal zeroes for some odd integer \( k>1 \).

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Updated on: 10-Oct-2022

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