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For a quadratic ax² + bx + c = 0 so we know sum of roots = -b /a product of roots = c/a let roots of the eq x² - x - k(2x- 1) ⇒ x² - x - 2kx +k ⇒ x² - (1 - 2k)x +k be x , y so given x+ y = 0 so (1 - 2k )/1 = 0 ⇒ k = 1/2

First of all you must know that any quadratic equation can be written as : Note: This is a very powerful tool. Do remember it. X^2 -sx+p -----------------(i) where s is the sum of its roots and p is their product. On comparing your equation , we get p=k s=2k+1 As we had only to talk about their sum so It is given that s=0 This means that 2k+1=0 2k=-1 k=-1/2.