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To determine the range of values for the variable k that ensures the quadratic equation 2x^2 + kx + 8 = 0 possesses distinct real roots, one must analyze the discriminant of the equation. The discriminant, denoted as D, is a crucial component in the study of quadratic equations, as it provides insight into the nature of the roots. For a quadratic equation of the form ax^2 + bx + c = 0, the discriminant is calculated using the formula D = b^2 – 4ac. In this case, a equals 2, b equals k, and c equals 8, leading to the expression D = k^2 – 4(2)(8).
To ensure that the roots of the equation are distinct and real, the discriminant must be greater than zero. Therefore, we set up the inequality k^2 – 64 > 0. This inequality can be factored into (k – 8)(k + 8) > 0. Solving this inequality involves identifying the critical points, which are k = 8 and k = -8. By analysing the intervals created by these critical points, one can determine the ranges of k that satisfy the condition for distinct real roots.
Consequently, the solution to the inequality reveals that k must be either greater than 8 or less than -8. Thus, the set of values for k that guarantees the quadratic equation 2x^2 + kx + 8 = 0 has distinct real roots is k < -8 or k > 8. For students seeking assistance in understanding these concepts, an A Level Maths Tutor can provide valuable guidance in mastering the intricacies of quadratic equations and their discriminants.
