\begin{tabular}{l} Use the graph \( f(x)=-x^{2}+8 x+20 \) to determine the values of: \\ \hline 2.2 .1 \\ \( p \) for which \( -x^{2}+8 x+p=0 \) will have equal roots. \end{tabular}

To find the value of \( p \) for which the quadratic equation \( -x^2 + 8x + p = 0 \) has equal roots, we need to examine the discriminant. The quadratic discriminant is given by the formula \( D = b^2 - 4ac \). For the equation to have equal roots, the discriminant must be equal to zero. Here, \( a = -1 \), \( b = 8 \), and \( c = p \). Plugging these values into the discriminant formula, we get: \[ D = 8^2 - 4(-1)(p) = 64 + 4p \] Setting the discriminant equal to zero for equal roots: \[ 64 + 4p = 0 \] Solving for \( p \): \[ 4p = -64 \\ p = -16 \] So, the value of \( p \) for which the equation will have equal roots is \( p = -16 \).