Given that alpha and beta are roots of the equation 3x^2-5=0 form the equation whose roots are 2alpha-1/beta, 2alpha-/alpha?

To form the equation whose roots are (2\alpha - \frac{1}{\beta}) and (2\alpha - \frac{1}{\alpha}), we first need to understand the original equation and its roots. However, there seems to be a typo in the problem statement. Let's correct it and proceed with the assumption that the original equation is (3x^2 - 5 = 0), and we want to find the equation whose roots are (2\alpha - \frac{1}{\beta}) and (2\alpha - \frac{1}{\alpha}).

Step 1: Solve the Original Equation for Alpha and Beta

Given the equation (3x^2 - 5 = 0), we solve for (x): [3x^2 = 5] [x^2 = \frac{5}{3}] [x = \pm\sqrt{\frac{5}{3}}]

Thus, (\alpha = \sqrt{\frac{5}{3}}) and (\beta = -\sqrt{\frac{5}{3}}).

Step 2: Calculate the New Roots

We need to calculate (2\alpha - \frac{1}{\beta}) and (2\alpha - \frac{1}{\alpha}).

For (2\alpha - \frac{1}{\beta}): [2\alpha - \frac{1}{\beta} = 2\sqrt{\frac{5}{3}} - \frac{1}{-\sqrt{\frac{5}{3}}} = 2\sqrt{\frac{5}{3}} + \frac{\sqrt{3}}{\sqrt{5}}]

For (2\alpha - \frac{1}{\alpha}): [2\alpha - \frac{1}{\alpha} = 2\sqrt{\frac{5}{3}} - \frac{1}{\sqrt{\frac{5}{3}}} = 2\sqrt{\frac{5}{3}} - \frac{\sqrt{3}}{\sqrt{5}}]

Step 3: Form the New Quadratic Equation

To form the quadratic equation whose roots are these new values, we use the sum and product of the roots.

Sum of the Roots: [S = \left(2\sqrt{\frac{5}{3}} + \frac{\sqrt{3}}{\sqrt{5}}\right) + \left(2\sqrt{\frac{5}{3}} - \frac{\sqrt{3}}{\sqrt{5}}\right) = 4\sqrt{\frac{5}{3}}]

Product of the Roots: [P = \left(2\sqrt{\frac{5}{3}} + \frac{\sqrt{3}}{\sqrt{5}}\right)\left(2\sqrt{\frac{5}{3}} - \frac{\sqrt{3}}{\sqrt{5}}\right) = \left(2\sqrt{\frac{5}{3}}\right)^2 - \left(\frac{\sqrt{3}}{\sqrt{5}}\right)^2] [P = \frac{20}{3} - \frac{3}{5} = \frac{100 - 9}{15} = \frac{91}{15}]

Step 4: Construct the Quadratic Equation

The general form of a quadratic equation is (x^2 - Sx + P = 0). Substituting (S) and (P): [x^2 - 4\sqrt{\frac{5}{3}}x + \frac{91}{15} = 0]

However, this equation does not simplify into a standard form with rational coefficients due to the irrational numbers involved. For practical purposes, we would typically seek equations with rational or easily manageable coefficients. The given transformation leads to complex expressions, and without further simplification or clarification on handling irrational coefficients, this is the form we arrive at.

Note: The problem as stated leads to complex expressions for the new roots, which might not be what was intended. If there was a mistake in interpreting the problem, please clarify or correct the problem statement.