Find the values of k for which the equation $k x^{2} + 2 x + 1 = 0$ has real and distinct roots.

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Solution

$Given: k x^{2} + 2 x + 1 = 0 Here, a = k, b = 2 and c = 1 Discriminant D is given by : D = (b^{2} - 4 a c) = {(2)}^{2} - 4 \times k \times 1 = 4 - 4 k If D > 0, t he roots of the equation will be real and distinct. \Rightarrow 4 - 4 k > 0 \Rightarrow 4 (1 - k) > 0 \Rightarrow (1 - k) > 0 \Rightarrow k < 1 Thus, the equation has real and distinct roots for all real values of k < 1 .$

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