\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 determine the value of \( p \) for which the equation \( -x^{2}+8x+p=0 \) has equal roots, we need to use the condition for equal roots in a quadratic equation. A quadratic equation \( ax^2 + bx + c = 0 \) has equal roots when its discriminant \( D \) is equal to zero. The discriminant is given by: \[ D = b^2 - 4ac \] In our case, the coefficients are: - \( a = -1 \) - \( b = 8 \) - \( c = p \) Substituting these values into the discriminant formula, we get: \[ D = 8^2 - 4(-1)(p) \] Now, we set the discriminant equal to zero for the roots to be equal: \[ 8^2 - 4(-1)(p) = 0 \] Calculating \( 8^2 \): \[ 64 + 4p = 0 \] Now, we can solve for \( p \): \[ 4p = -64 \] \[ p = -16 \] Thus, the value of \( p \) for which the equation \( -x^{2}+8x+p=0 \) will have equal roots is: \[ \boxed{-16} \]