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Question

Find the roots of each of the following equations, if they exist, by applying the quadratic formula:
a2b2x2(4b43a4)x12a2b2=0,a0 and b0

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Solution

a2b2x2(4b43a4)x12a2b2=0

Here A=a2b2
B=(4b43a4)
C=12a2b2

D=discriminant=B24AC
=[((4b43a4)24×[a2b2]×[12a2b2]
=[16b8+9a824a4b4]+48a4b4
=16b8+9a824a4b4
=(4b4)2+2×4b×3a2+(3a2)2
=(4b4+3a4)2[(a+b)2=a2+b2+2ab]

Now, let us use the quadratic formula

x=B±B24AC2A


x=B±D2A

x=((4b43a4))±(4b4+3a4)22a2b2

x=(4b43a4) ± 4b4+3a42a2b2


now taking positive sign,

x=4b43a4 + 4b4+3a42a2b2

x=8b42a2b2

x=4b2a2

taking negative value :

x=4b43a4 4b4+3a42a2b2

x=6a42a2b2

x=3a2b2


X=4b2a2,3a2b2


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