First, subtract #color(red)(7)# from each side of the equation to put the polynomial in standard form while keeping the equation balanced:
#k^2 + 8k + 11 - color(red)(7) = 7 - color(red)(7)#
#k^2 + 8k + 4 = 0#
We can now use the quadratic equation to solve this problem:
The quadratic formula states:
For #color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0#, the values of #x# which are the solutions to the equation are given by:
#x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))#
Substituting:
#color(red)(1)# for #color(red)(a)#
#color(blue)(8)# for #color(blue)(b)#
#color(green)(4)# for #color(green)(c)# gives:
#x = (-color(blue)(8) +- sqrt(color(blue)(8)^2 - (4 * color(red)(1) * color(green)(4))))/(2 * color(red)(1))#
#x = (-8 +- sqrt(64 - 16))/2#
#x = (-8 - sqrt(48))/2# and #x = (-8 + sqrt(48))/2#
#x = (-8 - sqrt(16 * 3))/2# and #x = (-8 + sqrt(16 * 3))/2#
#x = (-8 - sqrt(16)sqrt(3))/2# and #x = (-8 + sqrt(16)sqrt(3))/2#
#x = (-8 - 4sqrt(3))/2# and #x = (-8 + 4sqrt(3))/2#
#x = -8/2 - (4sqrt(3))/2# and #x = -8/2 + (4sqrt(3))/2#
#x = -4 - 2sqrt(3)# and #x = -4 + 2sqrt(3)#
