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)(4)# for #color(red)(a)#
#color(blue)(-8)# for #color(blue)(b)#
#color(green)(-177)# for #color(green)(c)# gives:
#n = (-color(blue)(-8) +- sqrt(color(blue)(-8)^2 - (4 * color(red)(4) * color(green)(-177))))/(2 * color(red)(4))#
#n = (-8 +- sqrt(64 - (16 * color(green)(-177))))/8#
#n = (-8 +- sqrt(64 - (-2832)))/8#
#n = (-8 +- sqrt(64 + 2832))/8#
#n = (-8 +- sqrt(2896))/8#
#n = (-8 +- sqrt(16 * 181))/8#
#n = (-8 +- sqrt(16)sqrt(181))/8#
#n = (-8 +- 4sqrt(181))/8#
#n = -8/8 +- (4sqrt(181))/8#
#n = -1 +- sqrt(181)/2#
The Solution SetIs: ##n = {-1 - sqrt(181)/2, -1 + sqrt(181)/2}#