D=b²-4ac
and root of quad eqn=b+√D/a and b-√D/a
so root is irrespective of the sign of D, as it is under root D (√D).
the sign of the root of quad eqn depends if b>√D (positive root) or b<√d (negative="" root)="" (negative="">

Understanding Quadratic Equations and Roots
A quadratic equation is typically represented as:
\[ ax^2 + bx + c = 0 \]
where \( a \), \( b \), and \( c \) are constants, and \( a \neq 0 \).

Discriminant Overview
The discriminant (\( D \)) of a quadratic equation is given by the formula:
\[ D = b^2 - 4ac \]
The value of the discriminant provides crucial information about the nature of the roots of the equation.

Roots and Their Nature
1. **Positive Roots**: For a quadratic equation to have two positive roots, both roots must be greater than zero.
2. **Discriminant Values**:
- **Positive Discriminant (\( D > 0 \))**: Indicates two distinct real roots.
- **Zero Discriminant (\( D = 0 \))**: Indicates one real double root.
- **Negative Discriminant (\( D < 0="" \))**:="" indicates="" no="" real="" roots="" (complex="" roots).="" />

Why Positive Roots Imply Positive Discriminant
- **Sum and Product of Roots**: According to Vieta’s formulas:
- The sum of the roots (\( r_1 + r_2 = -\frac{b}{a} \)) is positive if both roots are positive.
- The product of the roots (\( r_1 \times r_2 = \frac{c}{a} \)) is also positive.
- **Implication on Discriminant**:
- If \( b^2 - 4ac > 0 \), it ensures that the roots are real and distinct.
- For both roots to be positive, \( b \) must be negative or small enough that the quantity \( 4ac \) does not exceed \( b^2 \).

Conclusion
Thus, if both roots of a quadratic equation are positive, it logically follows that the discriminant must be positive, ensuring the existence of two distinct real roots. This relationship underscores the importance of the discriminant in understanding the characteristics of quadratic equations.