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Question
If a2 - 3a + 1 = 0, and a ≠ 0; find:
- `a + 1/a`
- `a^2 + 1/a^2`
Solution
(i) Consider the given equation
a2 - 3a + 1 = 0
Rewrite the given equation, we have
a2 + 1 = 3a
⇒ `[ a^2 + 1 ]/a = 3`
⇒ `[ a^2/a + 1/a ] = 3`
⇒ `[ a + 1/a ] = 3` ...(1)
(ii) We need to find `a^2 + 1/a^2`:
We know the identity, (a + b)2 = a2 + b2 + 2ab
∴ `(a + 1/a )^2 = a^2 + 1/a^2 + 2` ...(2)
From equation (1), we have,
`a + 1/a` = 3
Thus, equation (2), becomes,
⇒ `(3)^2 = a^2 + 1/a^2 + 2`
⇒ 9 = `a^2 + 1/a^2 + 2`
⇒ `a^2 + 1/a^2 = 7`
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