Form the quadratic equation, if the roots are
`(2)/(3)and(3)/(2)`

Answer

To form a quadratic equation given the roots \( \frac{2}{3} \) and \( \frac{3}{2} \), we can follow these steps:

### Step 1: Identify the Roots
The given roots are:
- \( \alpha = \frac{2}{3} \)
- \( \beta = \frac{3}{2} \)

### Step 2: Calculate the Sum of the Roots
The sum of the roots \( \alpha + \beta \) is calculated as follows:
\[ \alpha + \beta = \frac{2}{3} + \frac{3}{2} \]
To add these fractions, we need a common denominator. The least common multiple of 3 and 2 is 6.
\[ \frac{2}{3} = \frac{4}{6} \quad \text{and} \quad \frac{3}{2} = \frac{9}{6} \]
Now, adding them:
\[ \alpha + \beta = \frac{4}{6} + \frac{9}{6} = \frac{13}{6} \]

### Step 3: Calculate the Product of the Roots
The product of the roots \( \alpha \beta \) is calculated as follows:
\[ \alpha \beta = \frac{2}{3} \times \frac{3}{2} \]
Calculating this gives:
\[ \alpha \beta = \frac{2 \times 3}{3 \times 2} = 1 \]

### Step 4: Form the Quadratic Equation
Using the standard form of a quadratic equation, which is:
\[ x^2 - (\alpha + \beta)x + \alpha \beta = 0 \]
Substituting the values we found:
\[ x^2 - \frac{13}{6}x + 1 = 0 \]

### Step 5: Eliminate the Fraction
To eliminate the fraction, we can multiply the entire equation by 6:
\[ 6x^2 - 13x + 6 = 0 \]

### Final Quadratic Equation
Thus, the quadratic equation formed is:
\[ 6x^2 - 13x + 6 = 0 \]

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