Al Fown an oquasion with nots which exceed by 2 the roots of the quadratic \[ 2 x^{2}-(p-4) x-(2 p+1)=0 \] (a) Find the valuc of \( k \) of the range of values of \( k \) in each case below:

To solve the quadratic inequality associated with the equation \( 2 x^{2}-(p-4) x-(2 p+1)=0 \), we first need to find the roots of the quadratic using the quadratic formula, which gives us: \[ x = \frac{(p-4) \pm \sqrt{(p-4)^2 + 16p+8}}{4} \] Now, according to the problem, we need to find a value of \( k \) for which the corresponding quadratic expression exceeds its roots by 2. This translates into the condition that \( f(x) > k \) when \( x \) is in the interval between the roots. The transformation leads us to seek \( k \) in relation to the quadratic function itself, ensuring that \( k \) sets bounds that are consistently above the minimum value achieved between the roots. --- When working with quadratics, a common mistake is overlooking the relationship between the roots, coefficients, and the vertex. Always remember that for a parabolic graph, the vertex represents either a minimum or maximum point. Don’t hesitate to verify that your discriminant is non-negative to ensure real roots exist! This is key in determining the behavior of the function and setting feasible bounds for \( k \). Adding to this, a nifty problem-solving tip is to sketch the quadratic function before diving deep into calculations. This visual aid can help you intuitively grasp the intervals where the function meets or exceeds certain values, and thus derive the necessary conditions effectively!