Identify Sum and Product of Roots

Given roots are $\alpha = \frac{2}{3}$ and $\beta = \frac{3}{2}$. Calculate their sum and product.

$\alpha + \beta = \frac{2}{3} + \frac{3}{2} = \frac{4 + 9}{6} = \frac{13}{6}$

Calculate Product of Roots

Multiply the two roots together to find the constant term coefficient value.

$\alpha\beta = \left(\frac{2}{3}\right) \times \left(/\frac{3}{2}\right) = 1$

Substitute into Quadratic Equation Formula

Use the standard quadratic equation formula given by sum and product of roots.

$x^2 - \left(\frac{13}{6}\right)x + 1 = 0$

Simplify and Clear Denominators

Multiply the entire equation by $6$ to eliminate the fraction and obtain integer coefficients.

$6x^2 - 13x + 6 = 0$