#[−8(3m^2−2m)]^2−4(1+4m^2)(36m^2−48m−9)=0#
#[−8 color(blue)((3m^2−2m))]^2 =4(1+4m^2)(36m^2−48m−9)#
Note:
As #color(blue)((m))# is common to both the terms of the L.H.S (#3m^2# and #2m# )we take it out of the bracket.
#[−8 color(blue)(m) (3m−2)]^2 =4(1+4m^2) color(purple)((36m^2−48m−9)#
Note:
#color(purple)((3)# is common to all the terms within the second bracket of the
R.H.S (#36m^2#, #−48m# and #−9# ) so, we take it out of the bracket.
#[−8 m (3m−2)]^2 =4(1+4m^2) * color(purple)((3)) (12m^2−16m−3)#
#(−8 m)^2 (3m−2)^2 =4 xx color(purple)((3)) (1+4m^2) * (12m^2−16m−3)#
#cancel64m^2 color(green)((3m−2))^2 =cancel12 (1+4m^2) *(12m^2−16m−3)#
#16m^2 color(green)((3m−2))^2 =3 (1+4m^2) * (12m^2−16m−3)#
Applying the property: #color(green)((a-b)^2= a^2-2ab +b^2# to #color()((3m−2))^2#
#(16m^2) color(green)((9m^2-12m +4)) =3 (1+4m^2) * (12m^2−16m−3)#
#(16m^2) * (9m^2) + (16m^2) * ( -12m) +(16m^2) * (4) =3 (1+4m^2) xx (12m^2−16m−3)#
#144m^4 -192m^3 +64m^2 =3 (color(blue)(1)+color(purple)(4m^2)) xx (12m^2−16m−3)#
#144m^4 -192m^3 +64m^2 =3 [(color(blue)(1 * (12m^2) + 1 * (-16m) + 1 * (-3)) + color(purple)(4m^2 *(12m^2) + 4m^2 (-16m) + 4m^2 (-3)]#
#144m^4 -192m^3 +64m^2 =3 [(12m^2 -16m -3 +48m^4 -64m^3 -12m^2]#
#144m^4 -192m^3 +64m^2 =3 [(cancel12m^2 -16m -3 +48m^4 -64m^3 -cancel12m^2]#
#144m^4 -192m^3 +64m^2 =3 (-16m -3 +48m^4 -64m^3 )#
#144m^4 -192m^3 +64m^2 =3* (-16m ) + 3 * (-3) + 3 * (48m^4) +3 * (-64m^3 )#
#144m^4 -192m^3 +64m^2 = -48m -9 +144m^4 -192m^3#
#cancel144m^4 -cancel192m^3 +64m^2 = -48m -9 +cancel144m^4 -cancel192m^3#
#64m^2 = -48m -9 #
We arrive at quadratic equation:
#64m^2 + 48m +9 =0 #
The equation is of the form #color(blue)(am^2+bm+c=0# where:
#a=64, b=48, c=9#
The Discriminant is given by:
#Delta=b^2-4*a*c#
# = (48)^2-(4*64 * 9)#
# = 2304 -2304=0#
The solution is found using the formula
#x=(-b+-sqrtDelta)/(2*a)#
#x = ((-48)+-sqrt(0))/(2*64)= (-48 +-0)/128#
#x=-48/128#
#x=-0.375#
