First, expand the term in parenthesis by multiplying each term within the parenthesis by the term outside the parenthesis:
#color(red)(8n)(9n - 5) = 180#
#(color(red)(8n) xx 9n) - (color(red)(8n) xx 5) = 180#
#72n^2 - 40n = 180#
Next, subtract #color(red)(180)# from each side of the equation to put the equation in standard quadratic form:
#72n^2 - 40n - color(red)(180) = 180 - color(red)(180)#
#72n^2 - 40n - 180 = 0#
We can make this equation a little easier to work with by factoring out a common factor from each term on the left:
#4(18n^2 - 10n - 45) = 0#
#(4(18n^2 - 10n - 45))/color(red)(4) = 0/color(red)(4)#
#(color(red)(cancel(color(black)(4)))(18n^2 - 10n - 45))/cancel(color(red)(4)) = 0#
#18n^2 - 10n - 45 = 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)(-10)# for #color(blue)(b)#
#color(green)(-45)# for #color(green)(c)# gives:
#x = (-color(blue)(-10) +- sqrt(color(blue)(-10)^2 - (4 * color(red)(18) * color(green)(-45))))/(2 * color(red)(18))#
#x = (10 +- sqrt(100 - (-3240)))/36#
#x = (10 +- sqrt(100 + 3240))/36#
#x = (10 +- sqrt(3340))/36#
#x = (10 +- sqrt(4 * 835))/36#
#x = 10/36 +- (sqrt(4)sqrt(835))/36#
#x = 5/18 +- (2sqrt(835))/36#
#x = 5/18 +- sqrt(835)/18#
