If one root of the equation x^2 - kx + k = 0 exceeds the other by 2 3 , then which one of the following is a value of k ?

Step-by-step solution

Let the roots of the equation x^2 - kx + k = 0 be and . Given that one root exceeds the other by 2 3 , we have | - | = 2 3 . Squaring both sides, we get ( - )^2 = 12 . Using the identity ( - )^2 = ( + )^2 - 4 , we can substitute the sum and product of the roots. From the given quadratic equation, + = k and = k . Substituting these values, we get k^2 - 4k = 12 . k^2 - 4k - 12 = 0 (k - 6)(k + 2) = 0 Thus, k = 6 or k = -2 . Among the given options, 6 is a value of k . Answer: 6