[−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