Question

If \( 3x^2 - 5x + 1 = 0 \), then the value of \( x^2 + \frac{1}{9x^2} \) will be:

• A. \( \frac{19}{9} \)
• B. 2
• C. \( \frac{17}{3} \)
• D. \( \frac{31}{9} \)

Hint:

When working with quadratic equations and their roots, it is helpful to manipulate the equation by first solving for the roots and then applying the required expressions directly.

Answer 1

The Correct Option is A

The given equation is: \[ 3x^2 - 5x + 1 = 0 \] To find \( x^2 + \frac{1}{9x^2} \), we can first solve for \( x^2 \). From the quadratic equation: \[ x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(3)(1)}}{2(3)} = \frac{5 \pm \sqrt{25 - 12}}{6} = \frac{5 \pm \sqrt{13}}{6} \] Now, to solve for \( x^2 + \frac{1}{9x^2} \), we need to work with the value of \( x^2 \) and find the expression \( \frac{1}{9x^2} \). Using algebraic simplifications, the correct value of \( x^2 + \frac{1}{9x^2} \) evaluates to \( \frac{19}{9} \).