The values of $\lambda$ for which the equation, $x^2- x+\lambda =0 $ has distinct real roots lie in the interval(s)

For a quadratic equation to have distinct real roots, the discriminant must be strictly greater than zero. The discriminant is (-1)^2 - 4(1)(lambda) = 1 - 4 * lambda. Setting this greater than zero gives 1 - 4 * lambda > 0, which means lambda < 1/4. Therefore, the values of lambda lie in the interval from negative infinity to 1/4.

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