2. Solve the equation `6 ( X^2 + 1/X^2 ) - 25 ( X - 1/X ) + 12 = 0`
`6 * (X^2+1/X^2) + (-25) * (X-1/X) + (12) = 0`
Let `X - 1/X = m`
`=> (X - 1/X)^2 = m^2`
`=> X^2 + 1/X^2 - 2 = m^2`
`=> X^2 + 1/X^2 = m^2 + 2`
Substituting this values in the given equation, we get
`6(m^2 + 2) - 25m + 12 = 0`
`6m^2 - 25m + 24 = 0`
`=> (6m^2-25m+24) = 0`
`=> 6m^2-25m+24 = 0`
`=> (6m^2-25m+24) = 0`
`=> (6m^2-16m-9m+24) = 0`
`=> 2m(3m-8)+(-3)(3m-8) = 0`
`=> (2m-3)(3m-8) = 0`
`=> (2m-3) = 0" or "(3m-8) = 0`
`=> 2m = 3" or "3m = 8`
`=> m = 3/2" or "m = 8/3`
Now, `X - 1/X = 3/2`
`=> 2X^2 - 2 = 3X`
`=> 2X^2 - 3X - 2 = 0`
`=> (2X^2-3X-2) = 0`
`=> 2X^2-3X-2 = 0`
`=> (2X^2-3X-2) = 0`
`=> (2X^2+X-4X-2) = 0`
`=> X(2X+1)+(-2)(2X+1) = 0`
`=> (X-2)(2X+1) = 0`
`=> (X-2) = 0" or "(2X+1) = 0`
`=> X = 2" or "2X = -1`
`=> X = 2" or "X = -1/2`
Now, `X - 1/X = 8/3`
`=> 3X^2 - 3 = 8X`
`=> 3X^2 - 8X - 3 = 0`
`=> (3X^2-8X-3) = 0`
`=> 3X^2-8X-3 = 0`
`=> (3X^2-8X-3) = 0`
`=> (3X^2+X-9X-3) = 0`
`=> X(3X+1)+(-3)(3X+1) = 0`
`=> (X-3)(3X+1) = 0`
`=> (X-3) = 0" or "(3X+1) = 0`
`=> X = 3" or "3X = -1`
`=> X = 3" or "X = -1/3`
This material is intended as a summary. Use your textbook for detail explanation.
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