Question

solve a2b2x2-(4b4-3a4)x-12a2b2=0 using quadratic formula

Answer 1

a²b²x² - (4b⁴ - 3a⁴)x - 12a²b² = 0
⇒a²b²x² - 4b⁴x + 3a⁴x - 12a²b² = 0
⇒b²x(a²x - 4b²) + 3a²(a²x - 4b²) = 0
⇒(b²x + 3a²)(a²x - 4b²) = 0
⇒x = 4b²/a², -3a²/b²

now, using quadratic formula ,
x = {(4b⁴ - 3b⁴) ± √{(4b⁴ -3a⁴)²+48(a²b²)²}}/2a²b²
= {(4b⁴ - 3a⁴) ± √{(4b⁴ + 3a⁴)²}}/2a²b² [ use formula, (x-y)²+4xy=(x+y)²]
= {4b⁴ - 3a⁴ ± (4b⁴ + 3a⁴)}/2a²b²
= 4b²/a², -3a²/b²

Answer 2

a²b²x² - (4b⁴ - 3a⁴)x - 12a²b² = 0

Here A=a²b²
B= -(4b⁴ - 3a⁴)
C= -  12a²b²

D=discriminant =B²-4AC=
=[( -(4b⁴ - 3a⁴)² - 4 x [a²b²] x -  12a²b²]
=[16b⁸ + 9a⁸ -24a⁴b⁴]+ 48 a⁴ b⁴
=16b⁸ + 9a⁸+24 a⁴ b⁴
=(4b⁴)² +2x 4b x 3a ² + (3a²)²
=(4b⁴+3a⁴)²     [∵ (a+b)²=a²+b²+2ab]

Now, let us use quadratic formula 

X = {-B ± √(B²-4AC)}/2A
= -( -(4b⁴ - 3a⁴)  ±√ (4b⁴+3a⁴)² /2a²b²
={(4b⁴ - 3a⁴)  ± (4b⁴+3a⁴)} /2a²b²
now taking positive sign,
=[(4b⁴ - 3a⁴) + 4b⁴ +3a⁴]/2a²b²
=8b⁴/2a²b²
x=4b²/a²

taking negative value :
=[(4b⁴ - 3a⁴) - 4b⁴ -3a⁴]/2a²b²
=-6a⁴/2a²b²
= -3a²/b²

∴X=4b²/a² ,  -3a²/b²