First, factor the equation to find the roots:

#y=x^2-4x-5# Set #y=0# to find the roots of the equation.

#0=x^2-4x-5# Factor

#(x+1)(x-5)=0# Using the zero products property,

#(x+1)=0# and #(x-5)=0# so the roots are:

#x=-1, 5#

Since #a# of #ax^2+bx+c# for this equation is positive, it opens upward with a minimum value, which is below #0# because it has #2# roots. Since parabolas are symmetric, the axis of symmetry must be in the middle of the two roots:

A.o.S.#=((x_1+x_2)/2)# (adapted average formula)

A.o.S.#=(((-1)+5)/2)#

A.o.S.#=2#

Axis of Symmetry: #x=2#

The axis of symmetry will intersect the minimum of the parabola, so we can input #x=2# into the equation:

#y=x^2-4x-5#

#y=(2)^2-4(2)-5# Combining like terms:

#y=4-8-5# Combining like terms:

#y=-9#

Therefore, the minimum is #(2, -9)#