If x^2-3x+1=0 prove that x^2+1/x^2=7?

Question

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Answer 1

If x₁ and x₂ are the roots of the given equation, substituted into equation, they verify it.

x₁² - 3x₁ + 1 = 0 ---------------> (1)

x₂² - 3x₂ + 1 = 0 -------------- > (2)

We'll add (1) + (2):

x₁² + x₂² - 3(x₁ + x₂) + 2 = 0

Since we'll have to determine the sum of the squares, we'll move to the right side of the equal sign, the rest of the expression.

x₁² + x₂² = 3(x₁ + x₂) - 2

We'll apply Viete's relations for finding out  x₁ + x₂:

x₁ + x₂ = -b/a

x₁ + x₂ = 3

We'll substitute the sum of the roots into :

x₁² + x₂² = 3*3 - 2

x₁² + x₂² = 9 - 2

x₁² + x₂² = 7

It is obvious that 7 is a natural number, so the sum of the squares of the roots of the equation is a natural number.