Nature of Roots and Advanced Conditions

Nature of Roots and Advanced Conditions

This lesson covers the following key concepts:

• Nature of roots based on discriminant
• Real and distinct roots (D > 0)
• Real and equal roots (D = 0)
• Complex conjugate roots (D < 0)
• Rational roots condition (D = perfect square)
• Conditions for both roots positive, negative, opposite signs
• Relation between coefficients for specific root conditions

Important Formulas

• $$D > 0: \text{real and distinct roots}$$
• $$D = 0: \text{real and equal roots}$$
• $$D < 0: \text{complex conjugate roots}$$
• $$D > 0, D = k^2: \text{rational roots}$$
• $$\text{Both roots positive: } \alpha + \beta > 0, \alpha\beta > 0, D \geq 0$$
• $$\text{Both roots negative: } \alpha + \beta < 0, \alpha\beta > 0, D \geq 0$$
• $$\text{Opposite signs: } \alpha\beta < 0$$