If x + 1/x = 15, then find the value of (7x^2 - 9x + 7) / (x^2 - x + 1).

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Accepted Answer

Given: x + 1/x = 15

Step 1: Find x^2 + 1/x^2

Square both sides:
(x + 1/x)^2 = 15^2
x^2 + 2 + 1/x^2 = 225
x^2 + 1/x^2 = 225 - 2 = 223

Step 2: Express numerator and denominator in terms of x + 1/x

Numerator: 7x^2 - 9x + 7
Denominator: x^2 - x + 1

Rewrite numerator and denominator by grouping:

Numerator = 7x^2 + 7 - 9x = 7(x^2 + 1) - 9x
Denominator = x^2 + 1 - x

Step 3: Use substitution for x^2 + 1

Alternatively, divide numerator and denominator by x to simplify:

Numerator / x = 7x - 9 + 7/x
Denominator / x = x - 1 + 1/x

Let y = x + 1/x = 15
Then numerator / x = 7x + 7/x - 9 = 7(x + 1/x) - 9 = 7*15 - 9 = 105 - 9 = 96
Denominator / x = x + 1/x - 1 = 15 - 1 = 14

Therefore, (7x^2 - 9x + 7) / (x^2 - x + 1) = (Numerator / x) / (Denominator / x) = 96 / 14 = 48 / 7

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