First, put the equation in standard form:
#6x^2 + color(6x) - 36 = -6x + color(6x)#
#6x^2 + 6x - 36 = 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)(6)# for #color(red)(a)#
#color(blue)(6)# for #color(blue)(b)#
#color(green)(-36)# for #color(green)(c)# gives:
#x = (-color(blue)(6) +- sqrt(color(blue)(6)^2 - (4 * color(red)(6) * color(green)(-36))))/(2 * color(red)(6))#
#x = (-color(blue)(6) +- sqrt(36 - (-864)))/12#
#x = (-color(blue)(6) +- sqrt(36 + 864))/12#
#x = (-color(blue)(6) - sqrt(900))/12# and #x = (-color(blue)(6) + sqrt(900))/12#
#x = (-color(blue)(6) - 30)/12# and #x = (-color(blue)(6) + 30)/12#
#x = -36/12# and #x = 24/12#
#x = -3# and #x = 2#
Another way of solving this quadratic equation:
#6x^2+6x-36=0#
#(3x+9)(2x-4)=0#
#3x# must be -9 or #2x# must be 4 to end up with 0.
#3x=-9#
#x=-3#
#2x=4#
#x=2#
Therefor #x=-3# or #x=2#
