If `alpha and beta ` are the roots of the equation` 3x^(2) -4x+ 1 =0` ,find the equation whose roots are `(alpha^(2))/( beta) and ( beta^(2))/( alpha )`

Answer

To find the equation whose roots are \(\frac{\alpha^2}{\beta}\) and \(\frac{\beta^2}{\alpha}\), where \(\alpha\) and \(\beta\) are the roots of the equation \(3x^2 - 4x + 1 = 0\), we can follow these steps:

### Step 1: Find the sum and product of the roots \(\alpha\) and \(\beta\)

For a quadratic equation of the form \(ax^2 + bx + c = 0\):
- The sum of the roots \(\alpha + \beta = -\frac{b}{a}\)
- The product of the roots \(\alpha \beta = \frac{c}{a}\)

Here, \(a = 3\), \(b = -4\), and \(c = 1\).

Calculating the sum:
\[ \alpha + \beta = -\frac{-4}{3} = \frac{4}{3} \]

Calculating the product:
\[ \alpha \beta = \frac{1}{3} \]

### Step 2: Find the new roots \(\frac{\alpha^2}{\beta}\) and \(\frac{\beta^2}{\alpha}\)

The sum of the new roots can be calculated as:
\[ \frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha \beta} \]

Using the identity for the sum of cubes:
\[ \alpha^3 + \beta^3 = (\alpha + \beta)(\alpha^2 - \alpha\beta + \beta^2) \]
We can express \(\alpha^2 + \beta^2\) using the square of the sum of the roots:
\[ \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta = \left(\frac{4}{3}\right)^2 - 2\left(\frac{1}{3}\right) = \frac{16}{9} - \frac{2}{3} = \frac{16}{9} - \frac{6}{9} = \frac{10}{9} \]

Now substituting back:
\[ \alpha^3 + \beta^3 = \left(\frac{4}{3}\right)\left(\frac{10}{9} - \frac{1}{3}\right) = \left(\frac{4}{3}\right)\left(\frac{10}{9} - \frac{3}{9}\right) = \left(\frac{4}{3}\right)\left(\frac{7}{9}\right) = \frac{28}{27} \]

Thus, the sum of the new roots is:
\[ \frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha \beta} = \frac{\frac{28}{27}}{\frac{1}{3}} = \frac{28}{27} \cdot 3 = \frac{84}{27} = \frac{28}{9} \]

### Step 3: Find the product of the new roots

The product of the new roots is:
\[ \frac{\alpha^2}{\beta} \cdot \frac{\beta^2}{\alpha} = \frac{\alpha^2 \beta^2}{\alpha \beta} = \alpha \beta = \frac{1}{3} \]

### Step 4: Form the new quadratic equation

The new quadratic equation with roots \(\frac{\alpha^2}{\beta}\) and \(\frac{\beta^2}{\alpha}\) can be formed using the sum and product of the roots:
\[ x^2 - \left(\text{sum of roots}\right)x + \left(\text{product of roots}\right) = 0 \]
Substituting the values:
\[ x^2 - \frac{28}{9}x + \frac{1}{3} = 0 \]

To eliminate the fractions, multiply through by 9:
\[ 9x^2 - 28x + 3 = 0 \]

### Final Answer
Thus, the required equation is:
\[ \boxed{9x^2 - 28x + 3 = 0} \]