1. Solve the equation `1 (X^2 + 1/X^2) - 8 ( X + 1/X ) + 14 = 0`
`1 * (X^2+1/X^2) + (-8) * (X+1/X) + (14) = 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
`(m^2 - 2) - 8m + 14 = 0`
`m^2 - 8m + 12 = 0`
`=> (m^2-8m+12) = 0`
`=> m^2-8m+12 = 0`
`=> (m^2-8m+12) = 0`
`=> (m^2-2m-6m+12) = 0`
`=> m(m-2)+(-6)(m-2) = 0`
`=> (m-6)(m-2) = 0`
`=> (m-6) = 0" or "(m-2) = 0`
`=> m = 6" or "m = 2`
Now, `X + 1/X = 6`
`=> X^2 + 1 = 6X`
`=> X^2 - 6X + 1 = 0`
`=> (X^2-6X+1) = 0`
`=> X^2-6X+1 = 0`
Comparing the given equation with the standard quadratic equation `ax^2 + bx + c = 0,`
we get, `a = 1, b = -6, c = 1.`
`:. Delta = b^2 - 4ac`
` = (-6)^2 - 4 (1) (1)`
` = 36 - 4`
` = 32`
`:. sqrt(Delta) = sqrt(32) = 4 * sqrt(2)`
Now, `alpha = (-b + sqrt(Delta)) / (2a)`
` = (-(-6) + 4 * sqrt(2)) / (2 * 1)`
` = (6 + 4 * sqrt(2)) / 2`
` = 3 + 2 * sqrt(2)`
and, `beta = (-b - sqrt(Delta)) / (2a)`
` = (-(-6) - 4 * sqrt(2)) / (2 * 1)`
` = (6 - 4 * sqrt(2)) / 2`
` = 3 - 2 * sqrt(2)`
Now, `X + 1/X = 2`
`=> X^2 + 1 = 2X`
`=> X^2 - 2X + 1 = 0`
`=> (X^2-2X+1) = 0`
`=> X^2-2X+1 = 0`
`=> (X^2-2X+1) = 0`
`=> (X-1)(X-1) = 0`
`=> (X-1) = 0" or "(X-1) = 0`
`=> X = 1" or "X = 1`
This material is intended as a summary. Use your textbook for detail explanation.
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