- Identify the coefficients of the quadratic equation as \(a = 5\), \(b = -2\), and \(c = 6\).
- Calculate the discriminant \(D = (-2)^2 - 4(5)(6) = -116\), confirming the roots are complex.
- Use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to find the roots: \(x = \frac{2 \pm i\sqrt{116}}{10}\).
- Simplify the roots to obtain \(x = \frac{1 \pm i\sqrt{29}}{5}\), which matches option c. \(\boxed{c}\)
Explanation
- Analysis of the Quadratic Equation
The equation given is a standard quadratic equation of the form \(ax^2 + bx + c = 0\), where \(a = 5\), \(b = -2\), and \(c = 6\). To solve for \(x\), we use the quadratic formula:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]
This formula provides the values of \(x\) that satisfy the quadratic equation, regardless of whether the roots are real or complex.
- Calculating the Discriminant and Simplifying Roots
First, we calculate the discriminant \(D = b^2 - 4ac\), which helps us understand the nature of the roots:
\[D = (-2)^2 - 4 \times 5 \times 6\]
\[D = 4 - 120 = -116\]
Since the discriminant is less than zero (\(D < 0\)), the equation has two complex conjugate roots. We can express the square root of the discriminant as:
\[\sqrt{D} = \sqrt{-116} = i\sqrt{116} = i\sqrt{4 \times 29} = 2i\sqrt{29}\]
- Solving for x and Selecting the Correct Option
Now we substitute the values of \(a\), \(b\), and the calculated root expression into the quadratic formula:
\[x = \frac{-(-2) \pm 2i\sqrt{29}}{2 \times 5}\]
\[x = \frac{2 \pm 2i\sqrt{29}}{10}\]
By dividing the numerator and the denominator by their greatest common divisor, which is 2, we obtain:
\[x = \frac{1 \pm i\sqrt{29}}{5}\]
Comparing this result to the given options, it corresponds to option c.
Examples
Quadratic equations with complex roots are frequently encountered in electrical engineering when analyzing RLC circuits (Resistor-Inductor-Capacitor circuits). For instance, when the resistance is relatively small compared to the inductance and capacitance, the circuit can become "underdamped," resulting in oscillatory behavior. The complex roots of the characteristic equation represent the frequency of oscillation and the rate at which the signal decays, which is crucial for engineers designing filters or communication systems.