First, expand the squared term using this rule for multiplying quadratics:
#(color(red)(x) + color(blue)(y))^2 = color(red)(x)^2 + 2color(red)(x)color(blue)(y) + color(blue)(y)^2#
Substituting #color(red)(4n)# for #color(red)(x)# and #color(blue)(3)# for #color(blue)(y)# gives:
#(color(red)((4n)) + color(blue)(3))^2 - 11 = 0#
#color(red)((4n))^2 + (2 * color(red)(4n) * color(blue)(3)) + color(blue)(3)^2 - 11 = 0#
#16n^2 + 24n + 9 - 11 = 0#
#16n^2 + 24n - 2 = 0#
We can 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)(16)# for #color(red)(a)#
#color(blue)(24)# for #color(blue)(b)#
#color(green)(-2)# for #color(green)(c)# gives:
#n = (-color(blue)(24) +- sqrt(color(blue)(24)^2 - (4 * color(red)(16) * color(green)(-2))))/(2 * color(red)(16))#
#n = (-color(blue)(24) +- sqrt(color(blue)(576) - (-128)))/32#
#n = (-color(blue)(24) +- sqrt(color(blue)(576) + 128))/32#
#n = (-color(blue)(24) +- sqrt(704))/32#
#n = (-color(blue)(24) +- sqrt(64 * 11))/32#
#n = (-color(blue)(24) +- sqrt(64)sqrt(11))/32#
#n = (-color(blue)(24) +- 8sqrt(11))/32#
#n = (8(-color(blue)(3) +- sqrt(11)))/32#
#n = (color(red)(cancel(color(black)(8)))(-color(blue)(3) +- sqrt(11)))/(color(red)(cancel(color(black)(32)))4)#
#n = (-3 +- sqrt(11))/4#
