Find the value of the discriminant and identify the nature of the roots, a. \( D=4 \), the roots are real numbers and are equal. b. \( D=4 \), the roots are irrational numbers and are not equal. c. \( D=4 \), the roots are rational numbers but are not equal. d. \( D=4 \), the equation has no real roots.

To determine the nature of the roots of a quadratic equation, we use the discriminant, which is given by the formula: \[ D = b^2 - 4ac \] where \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). Let's analyze each case based on the given discriminant \( D = 4 \): a. \( D = 4 \), the roots are real numbers and are equal. For the roots to be real and equal, the discriminant must be zero. Since \( D = 4 \neq 0 \), this case is not correct. b. \( D = 4 \), the roots are irrational numbers and are not equal. For the roots to be irrational and not equal, the discriminant must be positive. Since \( D = 4 > 0 \), this case is correct. c. \( D = 4 \), the roots are rational numbers but are not equal. For the roots to be rational and not equal, the discriminant must be positive. Since \( D = 4 > 0 \), this case is correct. d. \( D = 4 \), the equation has no real roots. For the equation to have no real roots, the discriminant must be negative. Since \( D = 4 \neq 0 \), this case is not correct. Therefore, the correct answers are: b. \( D = 4 \), the roots are irrational numbers and are not equal. c. \( D = 4 \), the roots are rational numbers but are not equal.