Finding the Quadratic Equation from Roots

To find the quadratic equation given its roots, we use the relationship between the roots and the coefficients of the equation. If the roots of a quadratic equation are α and β, the equation can be written as: $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ or $x^2 - (\alpha + \beta)x + \alpha\beta = 0$

Identify Given Roots

The given roots are:

• Root 1: α = -2
• Root 2: β = 3

Calculate Sum and Product of Roots

Calculate the sum and product using the given roots:

• Sum of roots:
  $\alpha + \beta = -2 + 3 = 1$
• Product of roots:
  $\alpha \beta = (-2) \times 3 = -6$

Form the Quadratic Equation

Substitute the calculated sum and product into the standard equation format:

$x^2 - (\alpha + \beta)x + \alpha\beta = 0$
$x^2 - (1)x + (-6) = 0$
$x^2 - x - 6 = 0$

This equation matches option 2.