- The discriminant \(D = b^2 - 4ac\) determines the nature of the roots of a quadratic equation.
- Since \(D = -1 < 0\), the roots are given by \(\frac{-b \pm \sqrt{-1}}{2a} = \frac{-b \pm i}{2a}\).
- These roots are two distinct complex numbers.
- The correct choice is \(\boxed{\text{A}}\).
Explanation
- Analysis of the Quadratic Discriminant
The discriminant of a quadratic function \(q(x) = ax^2 + bx + c\) is given by the expression \(D = b^2 - 4ac\). This value is crucial because it directly determines the nature of the roots of the quadratic equation \(q(x) = 0\), as seen in the quadratic formula:
\[x = \frac{-b \pm \sqrt{D}}{2a}\]
In our specific case, we are given that the discriminant \(D = -1\). We need to determine the number and type of solutions for the equation \(q(x) = 0\).
- Determining the Nature of the Solutions
Since the discriminant \(D = -1\) is less than zero, the square root of the discriminant, \(\sqrt{D}\), involves the square root of a negative number. Specifically, \(\sqrt{-1} = i\), where \(i\) is the imaginary unit. Substituting this into the quadratic formula, we get:
\[x = \frac{-b \pm \sqrt{-1}}{2a} = \frac{-b \pm i}{2a}\]
This yields two distinct solutions:
\[x_1 = \frac{-b}{2a} + \frac{1}{2a}i \quad \text{and} \quad x_2 = \frac{-b}{2a} - \frac{1}{2a}i\]
Since \(i\) is present, these roots are complex numbers. Furthermore, because the \(\pm\) sign allows for two different values (provided \(a
eq 0\)), we have two distinct complex solutions.
- Conclusion
Based on our analysis, a quadratic equation with a discriminant of \(-1\) will always result in two distinct, conjugate complex solutions. Comparing this to the provided choices:
- A. 2 complex solutions
- B. 2 real solutions
- C. 1 complex solution
- D. 1 real solution
The correct option is A. Therefore, the quadratic \(q(x)\) has two complex solutions. Final answer: \(\boxed{\text{A}}\).
Examples
In electronics and signal processing, engineers often encounter systems described by differential equations whose characteristic equations have negative discriminants. This leads to complex roots, which represent oscillating behavior (like a damped harmonic oscillator). Knowing how to identify that a negative discriminant results in complex solutions allows engineers to predict that the system will exhibit underdamped oscillations rather than steady-state decay or instability.