If a quadratic equation has no real solutions, what type of solutions exist in the complex number system?

If a quadratic equation has no real solutions, what type of solutions exist in the complex number system?

• A. No solutions
• B. Two complex conjugate solutions
• C. One real solution
• D. Infinite solutions

Answer: (B)

A quadratic with real coefficients that has no real roots must have a negative discriminant, so its roots are not real numbers. In the complex number system you can write the roots as (-b ± sqrt(b^2 - 4ac)) / (2a). When the discriminant is negative, sqrt(b^2 - 4ac) becomes i√|b^2 - 4ac|, giving two roots of the form (-b ± i√|b^2 - 4ac|) / (2a). These two complex numbers come in a conjugate pair because the coefficients are real. So there are two complex conjugate solutions.

A quadratic with real coefficients that has no real roots must have a negative discriminant, so its roots are not real numbers. In the complex number system you can write the roots as (-b ± sqrt(b^2 - 4ac)) / (2a). When the discriminant is negative, sqrt(b^2 - 4ac) becomes i√|b^2 - 4ac|, giving two roots of the form (-b ± i√|b^2 - 4ac|) / (2a). These two complex numbers come in a conjugate pair because the coefficients are real. So there are two complex conjugate solutions.