What does it imply if the discriminant is a negative number?

When analyzing the discriminant of a quadratic equation, which can be expressed in the form ( ax^2 + bx + c = 0 ), the value of the discriminant is calculated as ( b^2 - 4ac ). The sign of the discriminant provides key information about the nature of the roots of the equation.

If the discriminant is a negative number, it indicates that there are no real solutions to the quadratic equation. Instead, the equation has two complex conjugate roots. This occurs because the square root of a negative number is imaginary, leading to complex solutions. Complex roots are typically represented in the form ( a + bi ), where ( a ) is the real part and ( bi ) is the imaginary part.

Thus, when the discriminant is negative, it directly implies that the quadratic equation has two complex roots, confirming that the correct choice is indeed the option referring to two complex roots. This outcome is fundamental in understanding the behavior of quadratic equations and their solutions in different scenarios.