Let and be the roots of the quadratic equation x^2 - 6x + k = 0 , where k Z . Suppose and represent the lengths of the two perpendicular legs of a right-angled triangle. If the hypotenuse H of this triangle satisfies the condition 3 2 H 5 , then the sum of all possible values of k is :

Step-by-step solution

Let the given quadratic equation be x^2 - 6x + k = 0 . The sum of the roots is + = 6 and the product of the roots is = k . Since and are the lengths of the perpendicular legs of a right-angled triangle, the length of the hypotenuse H is given by: H^2 = ^2 + ^2 We can express this in terms of the sum and product of the roots: H^2 = ( + )^2 - 2 = 36 - 2k We are given that 3 2 H 5 . Squaring all parts of the inequality, we get: 18 H^2 25 Substituting H^2 = 36 - 2k : 18 36 - 2k 25 -18 -2k -11 5.5 k 9 Since k is an inte