First, add #color(red)(6n)# to each side of the equation to put the equation in standard form while keeping the equation balanced:
#18n^2 - 30n + color(red)(6n) - 24 = -6n + color(red)(6n)#
#18n^2 + (-30 + color(red)(6))n - 24 = 0#
#18n^2 + (-24)n - 24 = 0#
#18n^2 - 24n - 24 = 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)(18)# for #color(red)(a)#
#color(blue)(-24)# for #color(blue)(b)#
#color(green)(-24)# for #color(green)(c)# gives:
#x = (-color(blue)(-24) +- sqrt(color(blue)(-24)^2 - (4 * color(red)(18) * color(green)(-24))))/(2 * color(red)(18))#
#x = (24 +- sqrt(576 - (72 * color(green)(-24))))/36#
#x = (24 +- sqrt(576 - (-1728)))/36#
#x = (24 +- sqrt(576 + 1728))/36#
#x = (24 +- sqrt(2304))/36#
#x = (24 +- 48)/36#
#x = (24 - 48)/36# and #x = (24 + 48)/36#
#x = -24/36# and #x = 72/36#
#x = -2/3# and #x = 2#
